# Particle Physics Planet

## May 27, 2017

## May 26, 2017

### Christian P. Robert - xi'an's og

While in Dublin last weekend, I found myself without a book to read and walking by and in a nice bookstore on Grafton Street, I discovered that Guy Gavriel Kay had published a book recently! Now, this was a terrific surprise as his Song for Arbonne was and remains one of my favourite books.

There are similarities in those two books in that they are both inspired by Mediterranean cultures and history, A Song for Arbonne being based upon the Late Medieval courts of Love in Occitany, while Children of Earth and Sky borrows to the century long feud between Venezia and the Ottoman empire, with Croatia stuck in-between. As acknowledged by the author, this novel stemmed from a visit to Croatia and the suggestion to tell the story of local bandits turned into heroes for fighting the Ottomans. Although I found unravelling the numerous borrowings from history and geography a wee bit tiresome, this is a quite enjoyable pseudo-historical novel. Except the plot is too predictable in having all its main characters crossing one another path with clockwise regularity. And all main women character eventually escaping the fate set upon them by highly patriarchal societies. A Song for Arbonne had more of a tension and urgency, or maybe made me care more for its central characters.

Filed under: Books, Kids, Travel Tagged: A Song for Arbonne, Children of Earth and Sky, Constantinople, Croatia, fantasy, Guy Gavriel Kay, historical novels, Prague, Venezia

### Emily Lakdawalla - The Planetary Society Blog

### Peter Coles - In the Dark

George Gershwin’s beautiful song *Summertime* has been recorded countless times in countless ways by countless artists, but if you’re expecting it to be performed as a restful lullaby, as it is normally played, you’ll probably be shocked. This version is a heartbreaking expression of pain and anguish performed by the great Albert Ayler, and it was recorded in Copenhagen in 1963.

P.S. The painting shown in the video is by Matisse….

Follow @telescoper### Clifford V. Johnson - Asymptotia

Here's a video glimpse (less than 1 min. long) of my working through designing the main character for the upcoming graphic short story I'm doing for an anthology to be published next year. (See here for more.) There's a clickable still on the right. I had started sketching her out on the subway a few days ago, and then finished some of the groundwork today on the bus, taking a snap at the end. From there I pulled it into ProCreate on the iPad pro, and then drew and painted more refined lines and strokes using an apple pencil. Faces are funny things... it isn't really until the final tweaks at the end that I was happy with the drawing. I was ready to abandon the whole thing all along, having decided that it was a failed drawing. So you never know. Always good to persist until the end... wherever that is. Last note: This drawing style is more detailed than I hope to use in the story. I will work out simpler versions of her for the story... I hope. Video below.

[...] Click to continue reading this post

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### Symmetrybreaking - Fermilab/SLAC

The Heavy Photon Search at Jefferson Lab is looking for a hypothetical particle from a hidden “dark sector.”

In 2015, a group of researchers installed a particle detector just half of a millimeter away from an extremely powerful electron beam. The detector could either start them on a new search for a hidden world of particles and forces called the “dark sector”—or its sensitive parts could burn up in the beam.

Earlier this month, scientists presented the results from that very first test run at the Heavy Photon Search collaboration meeting at the US Department of Energy’s Thomas Jefferson National Accelerator Facility. To the scientists’ delight, the experiment is working flawlessly.

Dark sector particles could be the long-sought components of dark matter, the mysterious form of matter thought to be five times more abundant in the universe than regular matter. To be specific, HPS is looking for a dark-sector version of the photon, the elementary “particle of light” that carries the fundamental electromagnetic force in the Standard Model of particle physics.

Analogously, the dark photon would be the carrier of a force between dark-sector particles. But unlike the regular photon, the dark photon would have mass. That’s why it’s also called the heavy photon.

To search for dark photons, the HPS experiment uses a very intense, nearly continuous beam of highly energetic electrons from Jefferson Lab’s CEBAF accelerator. When slammed into a tungsten target, the electrons radiate energy that could potentially produce the mystery particles. Dark photons are believed to quickly decay into pairs of electrons and their antiparticles, positrons, which leave tracks in the HPS detector.

“Dark photons would show up as an anomaly in our data—a very narrow bump on a smooth background from other processes that produce electron-positron pairs,” says Omar Moreno from SLAC National Accelerator Laboratory, who led the analysis of the first data and presented the results at the collaboration meeting.

The challenge is that, due to the large beam energy, the decay products are compressed very narrowly in beam direction. To catch them, the detector must be very close to the electron beam. But not too close—the smallest beam movements could make the beam swerve into the detector. Even if the beam doesn’t directly hit the HPS apparatus, electrons interacting in the target can scatter into the detector and cause unwanted signals.

The HPS team implemented a number of precautions to make sure their detector could handle the potentially destructive beam conditions. They installed and carefully aligned a system to intercept any large beam motions, made the detector’s support structure movable to bring the detector close to the beam and measure the exact beam position, and installed a feedback system that would shut the beam down if its motions were too large. They also placed their whole setup in vacuum because interactions of the beam with gas molecules would create too much background. Finally, they cooled the detector to negative 30 degrees Fahrenheit to reduce the effects of radiation damage. These measures allowed the team to operate their experiment so close to the beam.

“That’s maybe as close as anyone has ever come to such a particle beam,” says John Jaros, head of the HPS group at SLAC, which built the innermost part of the HPS detector, the Silicon Vertex Tracker. “So, it was fairly exciting when we gradually decreased the distance between the detector and the beam for the first time and saw that everything worked as planned. A large part of that success lies with the beautiful beams Jefferson Lab provided.”

SLAC’s Mathew Graham, who oversees the HPS analysis group, says, “In addition to figuring out if we can actually do the experiment, the first run also helped us understand the background signals in the experiment and develop the data analysis tools we need for our search for dark photons.”

So far, the team has seen no signs of dark photons. But to be fair, the data they analyzed came from just 1.7 days of accumulated running time. HPS collects data in short spurts when the CLAS experiment, which studies protons and neutrons using the same beam line, is not in use.

A second part of the analysis is still ongoing: The researchers are also closely inspecting the exact location, or vertex, from which an electron-positron pair emerges.

“If a dark photon lives long enough, it might make it out of the tungsten target where it was produced and travel some distance through the detector before it decays into an electron-positron pair,” Moreno says. The detector was specifically designed to observe such a signal.

Jefferson Lab has approved the HPS project for a total of 180 days of experimental time. Slowly but surely, HPS scientists are finding chances to use it.

### Tommaso Dorigo - Scientificblogging

As I explained in the previous post of this series, students in high schools of the Venice area have been asked to produce artistic works inspired by LHC physics research, and in particular the Higgs boson.

### Peter Coles - In the Dark

Since it’s a lovely sunny day in Cardiff – and already very warm – I thought I’d step outside the office of the Cardiff University Data Innovation Research Institute which is situated in the Trevithick Building and take a picture of our new sundial:

This flat sundial was installed by a company called Border Sundials and is designed very carefully to be as accurate as possible for the particular wall on which it is place. It’s also corrected for longitude.

However, I took the photograph at about 10.30am, and you’ll notice that it’s showing about 9.30. That’s because it hasn’t been corrected for British Summer Time so it’s offset by an hour. Moreover, a sundial always shows the local solar time rather than mean time which is shown on clocks. These differ because of (a) the inclination of the Earth’s orbit around the Sun relative to the equator and (b) the eccentricity of the Earth’s orbit around the Sun, which means that it does not move at a constant speed. The difference between mean time and solar time can be reconciled using the equation of time. The maximum correction is about 15 minutes, which is large enough to be seen on a sundial of this type. Often a graph of the equation of time is placed next to a sundial so one can do the correct oneself, but for some reason there isn’t one here.

The sundial adds quite a lot of interest to what otherwise is a featureless brick wall and we often notice people looking at it outside our office.

Follow @telescoper### Geraint Lewis - Cosmic Horizons

A little trawl of the internets reveals an awful lot of web pages discussing black holes, and discussions about spaghettification, firewalls, lost information, and many other things. Actually, a lot of the stuff out there on the web is nonsense, hand-waving, partly informed guesswork. And one of the questions that gets asked is "What would you see looking out into the universe?"

Some (incorrectly) say that you would never cross the event horizon, a significant mis-understanding of the coordinates of relativity. Other (incorrectly) conclude from this that you actually see the entire future history of the universe play out in front of your eyes.

What we have to remember, of course, is that relativity is a mathematical theory, and instead of hand waving, we can use mathematics to work out what we will see. And that's what I did.

I won't go through the details here, but it is based upon correctly calculating redshifts in relativity and conservation laws embodied in Killing vectors. But the result is an equation, an equation that looks like this

Here, r

_{s}is the radius from which you start to fall, r

_{e}is the radius at which the photon was emitted, and r

_{o}is the radius at which you receive the photon. On the left-hand-side is the ratio of the frequencies of the photon at the time of observation compared to emission. If this is bigger than one, then the photon is observed to have more energy than emitted, and the photon is blueshifted. If it is less than one, then it has less energy, and the photon is redshifted. Oh, and m is the mass of the black hole.

One can throw this lovely equation into python and plot it up. What do you get.

So, falling from a radius of 2.1m, we get

And falling from 3m

And from 5m

And 10m

and finally at 50m

In each of these, each line is a photon starting from different differences.

The key conclusion is that within the event horizon (r=2m) photons are generally seen to be redshifted, irrespective of where you start falling from. In fact in the last moment before you meet your ultimate end in the central singularity, the energy of the observed photon goes to zero and the outside universe is infinitely reshifted and vanishes from view.

How cool is that?

## May 25, 2017

### Christian P. Robert - xi'an's og

**A**n incomprehensible (and again double) Le Monde mathematical puzzle (despite requests to the authors! The details in brackets are mine.):

*A [non-circular] chain of 63 papers clips can be broken into sub-chains by freeing one clip [from both neighbours] at a time. At a given stage, considering the set of the lengths of these sub-chains, the collection of all possible sums of these lengths is a subset of {1,…,63}. What is the minimal number of steps to recover the entire set {1,…,63}? And w**hat is the maximal length L of a chain of paper clips that allows this recovery in 8 steps?**A tri-colored chain of 200 paper clips starts with a red, a blue and a green clip. Removing one clip every four clips produces a chain of 50 removed clips identical to the chain of 50 first clips of the original chain and a chain of remaining 150 clips identical to the 150 first clips of the original chain. Deduce the number of green, red, and blue clips.*

**T**he first question can be easily tackled by random exploration. Pick one number at random between 1 and 63, and keep picking attached clips until the set of sums is {1,…,63}. For instance,

rebreak0] sumz=cumsum(sample(difz)) for (t in 1:1e3) sumz=unique(c(sumz,cumsum(sample(difz)))) if (length(sumz)<63) brkz=rebreak(sort(c(brkz,sample((1:63)[-brkz],1)))) return(brkz)}

where I used sampling to find the set of all possible partial sums. Which leads to a solution with three steps, at positions 5, 22, and 31. This sounds impossibly small but the corresponding lengths are

1 1 1 4 8 16 32

from which one can indeed recover by summation all numbers till 63=2⁶-1. From there, a solution in 8 steps can be found by directly considering the lengths

1 1 1 1 1 1 1 1 9 18=9+8 36=18+17+1 72 144 288 576 1152 2303

whose total sum is 4607. And with breaks

10 29 66 139 284 573 1150 2303

The second puzzle is completely independent. Running another R code reproducing the constraints leads to

tromcol=function(N=200){ vale=rep(0,N) vale[1:3]=1:3 while (min(vale)==0){ vale[4*(1:50)]=vale[1:50] vale[-(4*(1:50))]=vale[1:150]} return(c(sum(vale==1),sum(vale==2),sum(vale==3)))}

and to 120 red clips, 46 blue clips and 34 green clips.

Filed under: Books, Kids Tagged: competition, Le Monde, mathematical puzzle, prime numbers, rank statistics

### Peter Coles - In the Dark

This amazing closeup image is of the North polar region of Jupiter. It was taken by NASA’s Juno spacecraft. Here’s a wider view:

I think it will take scientists quite some time to figure out what is going on in all those complex vortex structures!

In the meantime, though, I think these picture and the others that have been released can be enjoyed as a work of art! As a matter of fact reminds me of van Gogh’s Starry Night.*..*

### Clifford V. Johnson - Asymptotia

So, this is what the early stage of the graphic short story laying out process looks like. For me. I actually do it old school with pencil and paper, *and actual laying out*. You can click for a larger view but I've blurred out some bits - because spoilers.

So...20 pages works nicely. 16? Hmmmm...

-cvj Click to continue reading this post

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### Clifford V. Johnson - Asymptotia

Well, *that* was fun. And the NPR West studios in Culver City are fantastic.

I'll let you know when the piece, about science consulting for the entertainment industry, appears. Unless I really made a pig's ear of the interview in which case I may well forget to post it. ;)

-cvj Click to continue reading this post

The post At NPR West appeared first on Asymptotia.

### Peter Coles - In the Dark

I came across a paper on the arXiv yesterday with the title `*Why do we find ourselves around a yellow star instead of a red star?’. * Here’s the abstract:

M-dwarf stars are more abundant than G-dwarf stars, so our position as observers on a planet orbiting a G-dwarf raises questions about the suitability of other stellar types for supporting life. If we consider ourselves as typical, in the anthropic sense that our environment is probably a typical one for conscious observers, then we are led to the conclusion that planets orbiting in the habitable zone of G-dwarf stars should be the best place for conscious life to develop. But such a conclusion neglects the possibility that K-dwarfs or M-dwarfs could provide more numerous sites for life to develop, both now and in the future. In this paper we analyze this problem through Bayesian inference to demonstrate that our occurrence around a G-dwarf might be a slight statistical anomaly, but only the sort of chance event that we expect to occur regularly. Even if M-dwarfs provide more numerous habitable planets today and in the future, we still expect mid G- to early K-dwarfs stars to be the most likely place for observers like ourselves. This suggests that observers with similar cognitive capabilities as us are most likely to be found at the present time and place, rather than in the future or around much smaller stars.

Athough astrobiology is not really my province, I was intrigued enough to read on, until I came to the following paragraph in which the authors attempt to explain how Bayesian Inference works:

We approach this problem through the framework of Bayesian inference. As an example, consider a fair coin that is tossed three times in a row. Suppose that all three tosses turn up Heads. Can we conclude from this experiment that the coin must be weighted? In fact, we can still maintain our hypothesis that the coin is fair because the chances of getting three Heads in a row is 1/8. Many events with a probability of 1/8 occur every day, and so we should not be concerned about an event like this indicating that our initial assumptions are flawed. However, if we were to flip the same coin 70 times in a row with all 70 turning up Heads, we would readily conclude that the experiment is fixed. This is because the probability of flipping 70 Heads in a row is about 10

^{-22}, which is an exceedingly unlikely event that has probably never happened in the history of the universe. This

informal description of Bayesian inference provides a way to assess the probability of a hypothesis in light of new evidence.

Obviously I agree with the statement right at the end that `Bayesian inference provides a way to assess the probability of a hypothesis in light of new evidence’. That’s certainly what Bayesian inference does, but this `informal description’ is really a frequentist rather than a Bayesian argument, in that it only mentions the probability of given outcomes not the probability of different hypotheses…

Anyway, I was so unconvinced by this description’ that I stopped reading at that point and went and did something else. Since I didn’t finish the paper I won’t comment on the conclusions, although I am more than usually sceptical. You might disagree of course, so read the paper yourself and form your own opinion! For me, it goes in the file marked Bad Statistics!

Follow @telescoper### Emily Lakdawalla - The Planetary Society Blog

### The n-Category Cafe

One of the observations that launched homotopy type theory is that the rule of identity-elimination in Martin-Löf’s identity types automatically generates the structure of an $<semantics>\mathrm{\infty}<annotation\; encoding="application/x-tex">\backslash infty</annotation></semantics>$-groupoid. In this way, homotopy type theory can be viewed as a “synthetic theory of $<semantics>\mathrm{\infty}<annotation\; encoding="application/x-tex">\backslash infty</annotation></semantics>$-groupoids.”

It is natural to ask whether there is a similar *directed* type theory that describes a “synthetic theory of $<semantics>(\mathrm{\infty},1)<annotation\; encoding="application/x-tex">(\backslash infty,1)</annotation></semantics>$-categories”
(or even higher categories). Interpreting types directly as (higher) categories runs into various problems, such as the fact that not all maps between categories are exponentiable (so that not all $<semantics>\prod <annotation\; encoding="application/x-tex">\backslash prod</annotation></semantics>$-types exist), and that there are numerous different kinds of “fibrations” given the various possible functorialities and dimensions of categories appearing as fibers. The 2-dimensional directed type theory of Licata and Harper has semantics in 1-categories, with a syntax that distinguishes between co- and contra-variant dependencies; but since the 1-categorical structure is “put in by hand”, it’s not especially synthetic and doesn’t generalize well to higher categories.

An alternative approach was independently suggested by Mike and by Joyal, motivated by the model of homotopy type theory in the category of Reedy fibrant simplicial spaces, which contains as a full subcategory the $<semantics>\mathrm{\infty}<annotation\; encoding="application/x-tex">\backslash infty</annotation></semantics>$-cosmos of complete Segal spaces, which we call *Rezk spaces*. It is not possible to model ordinary homotopy type theory directly in the Rezk model structure, which is not right proper, but we can model it in the Reedy model structure and then identify internally some “types with composition,” which correspond to Segal spaces, and “types with composition and univalence,” which correspond to the Rezk spaces.

Almost five years later, we are finally developing this approach in more detail. In a new paper now available on the arXiv, Mike and I give definitions of *Segal* and *Rezk types* motivated by these semantics, and demonstrate that these simple definitions suffice to develop the synthetic theory of $<semantics>(\mathrm{\infty},1)<annotation\; encoding="application/x-tex">(\backslash infty,1)</annotation></semantics>$-categories. So far this includes functors, natural transformations, co- and contravariant type families with discrete fibers ($<semantics>\mathrm{\infty}<annotation\; encoding="application/x-tex">\backslash infty</annotation></semantics>$-groupoids), the Yoneda lemma (including a “dependent” Yoneda lemma that looks like “directed identity-elimination”), and the theory of coherent adjunctions.

## Cofibrations and extension types

One of the reasons this took so long to happen is that it required a technical innovation to become feasible. To develop the synthetic theory of Segal and Rezk types, we need to detect the semantic structure of the simplicial spaces model internally, and it seems that the best way to do this is to axiomatize the presence of a *strict interval* $<semantics>2<annotation\; encoding="application/x-tex">2</annotation></semantics>$ (a totally ordered set with distinct top and bottom elements). This is the geometric theory of which simplicial sets are the classifying topos (and of which simplicial spaces are the classifying $<semantics>(\mathrm{\infty},1)<annotation\; encoding="application/x-tex">(\backslash infty,1)</annotation></semantics>$-topos). We can then define an *arrow* in a type $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ to be a function $<semantics>2\to A<annotation\; encoding="application/x-tex">2\backslash to\; A</annotation></semantics>$.

However, often we want to talk about arrows with specified source and target. We can of course define the type $<semantics>{\mathrm{hom}}_{A}(x,y)<annotation\; encoding="application/x-tex">\backslash hom\_A(x,y)</annotation></semantics>$ of such arrows to be $<semantics>{\sum}_{f:2\to A}(f(0)=x)\times (f(1)=y)<annotation\; encoding="application/x-tex">\backslash sum\_\{f:2\backslash to\; A\}\; (f(0)=x)\backslash times\; (f(1)=y)</annotation></semantics>$, but since we are in homotopy type theory, the equalities $<semantics>f0=x<annotation\; encoding="application/x-tex">f0=x</annotation></semantics>$ and $<semantics>f1=y<annotation\; encoding="application/x-tex">f1=y</annotation></semantics>$ are *data*, i.e. homotopical paths, that have to be carried around everywhere. When we start talking about 2-simplices and 3-simplices with specified boundaries as well, the complexity becomes unmanageable.

The innovation that solves this problem is to introduce a notion of *cofibration* in type theory, with a corresponding type of *extensions*. If $<semantics>i:A\to B<annotation\; encoding="application/x-tex">i:A\backslash to\; B</annotation></semantics>$ is a cofibration and $<semantics>X:B\to \mathcal{U}<annotation\; encoding="application/x-tex">X:B\backslash to\; \backslash mathcal\{U\}</annotation></semantics>$ is a type family dependent on $<semantics>B<annotation\; encoding="application/x-tex">B</annotation></semantics>$, while $<semantics>f:{\prod}_{a:A}X(i(a))<annotation\; encoding="application/x-tex">f:\backslash prod\_\{a:A\}\; X(i(a))</annotation></semantics>$ is a section of $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$ over $<semantics>i<annotation\; encoding="application/x-tex">i</annotation></semantics>$, then we introduce an **extension type** $<semantics>\u27e8{\prod}_{b:B}X(b){\mid}_{f}^{i}\u27e9<annotation\; encoding="application/x-tex">\backslash langle\; \backslash prod\_\{b:B\}\; X(b)\; \backslash mid^i\_f\backslash rangle</annotation></semantics>$ consisting of “those dependent functions $<semantics>g:{\prod}_{b:B}X(b)<annotation\; encoding="application/x-tex">g:\backslash prod\_\{b:B\}\; X(b)</annotation></semantics>$ such that $<semantics>g(i(a))\equiv f(a)<annotation\; encoding="application/x-tex">g(i(a))\; \backslash equiv\; f(a)</annotation></semantics>$ — note the strict judgmental equality! — for any $<semantics>a:A<annotation\; encoding="application/x-tex">a:A</annotation></semantics>$”. This is modeled semantically by a “Leibniz” or “pullback-corner” map. In particular, we can define $<semantics>{\mathrm{hom}}_{A}(x,y)=\u27e8{\prod}_{t:2}A{\mid}_{[x,y]}^{0,1}\u27e9<annotation\; encoding="application/x-tex">\backslash hom\_A(x,y)\; =\; \backslash langle\; \backslash prod\_\{t:2\}\; A\; \backslash mid^\{0,1\}\_\{[x,y]\}\; \backslash rangle</annotation></semantics>$, the type of functions $<semantics>f:2\to A<annotation\; encoding="application/x-tex">f:2\backslash to\; A</annotation></semantics>$ such that $<semantics>f(0)\equiv x<annotation\; encoding="application/x-tex">f(0)\backslash equiv\; x</annotation></semantics>$ and $<semantics>f(1)\equiv y<annotation\; encoding="application/x-tex">f(1)\; \backslash equiv\; y</annotation></semantics>$ strictly, and so on for higher simplices.

General extension types along cofibrations were first considered by Mike and Peter Lumsdaine for a different purpose. In addition to the pullback-corner semantics, they are inspired by the path-types of cubical type theory, which replace the inductively specified identity types of ordinary homotopy type theory with a similar sort of restricted function-type out of the cubical interval. Our paper introduces a general notion of “type theory with shapes” and extension types that includes the basic setup of cubical type theory as well as our simplicial type theory, along with potential generalizations to Joyal’s “disks” for a synthetic theory of $<semantics>(\mathrm{\infty},n)<annotation\; encoding="application/x-tex">(\backslash infty,n)</annotation></semantics>$-categories.

## Simplices in the theory of a strict interval

In simplicial type theory, the cofibrations are the “inclusions of shapes” generated by the coherent theory of a strict interval, which is axiomatized by the interval $<semantics>2<annotation\; encoding="application/x-tex">2</annotation></semantics>$, top and bottom elements $<semantics>0,1:2<annotation\; encoding="application/x-tex">0,1\; :\; 2</annotation></semantics>$, and an inequality relation $<semantics>\le <annotation\; encoding="application/x-tex">\backslash le</annotation></semantics>$ satisfying the strict interval axioms.

Simplices can then be defined as

$$<semantics>{\Delta}^{n}:=\{\u27e8{t}_{1},\dots ,{t}_{n}\u27e9\mid {t}_{n}\le {t}_{n-1}\cdots {t}_{2}\le {t}_{1}\}<annotation\; encoding="application/x-tex">\; \backslash Delta^n\; :=\; \backslash \{\; \backslash langle\; t\_1,\backslash ldots,\; t\_n\backslash rangle\; \backslash mid\; t\_n\; \backslash leq\; t\_\{n-1\}\; \backslash cdots\; t\_2\; \backslash leq\; t\_1\; \backslash \}\; </annotation></semantics>$$

Note that the 1-simplex $<semantics>{\Delta}^{1}<annotation\; encoding="application/x-tex">\backslash Delta^1</annotation></semantics>$ agrees with the interval $<semantics>2<annotation\; encoding="application/x-tex">2</annotation></semantics>$.

Boundaries, e.g. of the 2-simplex, can be defined similarly $$<semantics>\partial {\Delta}^{2}:=\{\u27e8{t}_{1},{t}_{2}\u27e9:2\times 2\mid (0\equiv {t}_{2}\le {t}_{1})\vee ({t}_{2}\equiv {t}_{1})\vee ({t}_{2}\le {t}_{1}\equiv 1)\}<annotation\; encoding="application/x-tex">\; \backslash partial\backslash Delta^2\; :=\backslash \{\&\#10216;t\_1,t\_2\&\#10217;:\; 2\; \backslash times\; 2\; \backslash mid\; (0\; \backslash equiv\; t\_2\; \backslash leq\; t\_1)\; \backslash vee\; (t\_2\; \backslash equiv\; t\_1)\; \backslash vee\; (t\_2\; \backslash leq\; t\_1\; \backslash equiv\; 1)\backslash \}\; </annotation></semantics>$$ making the inclusion of the boundary of a 2-simplex into a cofibration.

## Segal types

For any type $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ with terms $<semantics>x,y:A<annotation\; encoding="application/x-tex">x,y\; :\; A</annotation></semantics>$ define

$$<semantics>{\mathrm{hom}}_{A}(x,y):=\u27e82\to A{\mid}_{[x,y]}^{\partial {\Delta}^{1}}\u27e9<annotation\; encoding="application/x-tex">\; hom\_A(x,y)\; :=\; \backslash langle\; 2\; \backslash to\; A\; \backslash mid^\{\backslash partial\backslash Delta^1\}\_\{\; [x,y]\}\; \backslash rangle\; </annotation></semantics>$$

That is, a term $<semantics>f:{\mathrm{hom}}_{A}(x,y)<annotation\; encoding="application/x-tex">f\; :\; hom\_A(x,y)</annotation></semantics>$, which we call an **arrow** from $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ to $<semantics>y<annotation\; encoding="application/x-tex">y</annotation></semantics>$ in $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$, is a function $<semantics>f:2\to A<annotation\; encoding="application/x-tex">f:\; 2\; \backslash to\; A</annotation></semantics>$ so that $<semantics>f(0)\equiv x<annotation\; encoding="application/x-tex">f(0)\; \backslash equiv\; x</annotation></semantics>$ and $<semantics>f(1)\equiv y<annotation\; encoding="application/x-tex">f(1)\; \backslash equiv\; y</annotation></semantics>$. For $<semantics>f:{\mathrm{hom}}_{A}(x,y)<annotation\; encoding="application/x-tex">f\; :\; hom\_A(x,y)</annotation></semantics>$, $<semantics>g:{\mathrm{hom}}_{A}(y,z)<annotation\; encoding="application/x-tex">g\; :\; hom\_A(y,z)</annotation></semantics>$, and $<semantics>h:{\mathrm{hom}}_{A}(x,z)<annotation\; encoding="application/x-tex">h\; :\; hom\_A(x,z)</annotation></semantics>$, a similar extension type

$$<semantics>{\mathrm{hom}}_{A}(x,y,z,f,g,h):=\u27e8{\Delta}^{2}\to A{\mid}_{[x,y,z,f,g,h]}^{\partial {\Delta}^{2}}\u27e9<annotation\; encoding="application/x-tex">\; hom\_A(x,y,z,f,g,h)\; :=\; \backslash langle\; \backslash Delta^2\; \backslash to\; A\; \backslash mid^\{\backslash partial\backslash Delta^2\}\_\{[x,y,z,f,g,h]\}\backslash rangle\; </annotation></semantics>$$

has terms that we interpret as witnesses that $<semantics>h<annotation\; encoding="application/x-tex">h</annotation></semantics>$ is the composite of $<semantics>f<annotation\; encoding="application/x-tex">f</annotation></semantics>$ and $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$. We define a **Segal type** to be a type in which any composable pair of arrows admits a unique (composite, witness) pair. In homotopy type theory, this may be expressed by saying that $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is **Segal** if and only if for all $<semantics>f:{\mathrm{hom}}_{A}(x,y)<annotation\; encoding="application/x-tex">f\; :\; hom\_A(x,y)</annotation></semantics>$ and $<semantics>g:{\mathrm{hom}}_{A}(y,z)<annotation\; encoding="application/x-tex">g\; :\; hom\_A(y,z)</annotation></semantics>$ the type

$$<semantics>\sum _{h:{\mathrm{hom}}_{A}(x,z)}{\mathrm{hom}}_{A}(x,y,z,f,g,h)<annotation\; encoding="application/x-tex">\; \backslash sum\_\{h\; :\; hom\_A(x,z)\}\; hom\_A(x,y,z,f,g,h)\; </annotation></semantics>$$

is contractible. A contractible type is in particular inhabited, and an inhabitant in this case defines a term $<semantics>g\circ f:{\mathrm{hom}}_{A}(x,z)<annotation\; encoding="application/x-tex">g\; \backslash circ\; f\; :\; hom\_A(x,z)</annotation></semantics>$ that we refer to as **the composite** of $<semantics>f<annotation\; encoding="application/x-tex">f</annotation></semantics>$ and $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$, together with a 2-simplex witness $<semantics>\mathrm{comp}(g,f):{\mathrm{hom}}_{A}(x,y,z,f,g,g\circ f)<annotation\; encoding="application/x-tex">comp(g,f)\; :\; hom\_A(x,y,z,f,g,g\; \backslash circ\; f)</annotation></semantics>$.

Somewhat surprisingly, this single contractibility condition characterizing Segal types in fact ensures coherent categorical structure at all dimensions. The reason is that if $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is Segal, then the type $<semantics>X\to A<annotation\; encoding="application/x-tex">X\; \backslash to\; A</annotation></semantics>$ is also Segal for any type or shape $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$. For instance, applying this result in the case $<semantics>X=2<annotation\; encoding="application/x-tex">X=2</annotation></semantics>$ allows us to prove that the composition operation in any Segal type is associative. In an appendix we prove a conjecture of Joyal that in the simplical spaces model this condition really does characterize exactly the Segal spaces, as usually defined.

## Discrete types

An example of a Segal type is a **discrete type**, which is one for which the map

$$<semantics>\mathrm{idtoarr}:\prod _{x,y:A}(x{=}_{A}y)\to {\mathrm{hom}}_{A}(x,y)<annotation\; encoding="application/x-tex">\; idtoarr\; :\; \backslash prod\_\{x,y:\; A\}\; (x=\_A\; y)\; \backslash to\; hom\_A(x,y)\; </annotation></semantics>$$

defined by identity elimination by sending the reflexivity term to the identity arrow, is an equivalence. In a discrete type, the $<semantics>\mathrm{\infty}<annotation\; encoding="application/x-tex">\backslash infty</annotation></semantics>$-groupoid structure encoded by the identity types and equivalent to the $<semantics>(\mathrm{\infty},1)<annotation\; encoding="application/x-tex">(\backslash infty,1)</annotation></semantics>$-category structure encoded by the hom types. More precisely, a type $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is discrete if and only if it is Segal, as well as *Rezk-complete* (in the sense to be defined later on), and moreover “every arrow is an isomorphism”.

## The dependent Yoneda lemma

If $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ and $<semantics>B<annotation\; encoding="application/x-tex">B</annotation></semantics>$ are Segal types, then any function $<semantics>f:A\to B<annotation\; encoding="application/x-tex">f:A\backslash to\; B</annotation></semantics>$ is automatically a “functor”, since by composition it preserves 2-simplices and hence witnesses of composition. However, not every type family $<semantics>C:A\to \mathcal{U}<annotation\; encoding="application/x-tex">C:A\backslash to\; \backslash mathcal\{U\}</annotation></semantics>$ is necessarily functorial; in particular, the universe $<semantics>\mathcal{U}<annotation\; encoding="application/x-tex">\backslash mathcal\{U\}</annotation></semantics>$ is not Segal — its hom-types $<semantics>{\mathrm{hom}}_{\mathcal{U}}(X,Y)<annotation\; encoding="application/x-tex">\backslash hom\_\{\backslash mathcal\{U\}\}(X,Y)</annotation></semantics>$ consist intuitively of “spans and higher spans”. We say that $<semantics>C:A\to \mathcal{U}<annotation\; encoding="application/x-tex">C:A\backslash to\; \backslash mathcal\{U\}</annotation></semantics>$ is **covariant** if for any $<semantics>f:{\mathrm{hom}}_{A}(x,y)<annotation\; encoding="application/x-tex">f:\backslash hom\_A(x,y)</annotation></semantics>$ and $<semantics>u:C(x)<annotation\; encoding="application/x-tex">u:C(x)</annotation></semantics>$, the type

$$<semantics>\sum _{v:C(y)}\u27e8\prod _{t:2}C(f(t)){\mid}_{[u,v]}^{\partial {\Delta}^{1}}\u27e9<annotation\; encoding="application/x-tex">\; \backslash sum\_\{v:C(y)\}\; \backslash langle\; \backslash prod\_\{t:2\}\; C(f(t))\; \backslash mid^\{\backslash partial\backslash Delta^1\}\_\{[u,v]\}\backslash rangle\; </annotation></semantics>$$

of “liftings of $<semantics>f<annotation\; encoding="application/x-tex">f</annotation></semantics>$ starting at $<semantics>u<annotation\; encoding="application/x-tex">u</annotation></semantics>$” is contractible. An inhabitant of this type consists of a point $<semantics>{f}_{*}(u):C(y)<annotation\; encoding="application/x-tex">f\_\backslash ast(u):C(y)</annotation></semantics>$, which we call the *(covariant) transport of $<semantics>u<annotation\; encoding="application/x-tex">u</annotation></semantics>$ along $<semantics>f<annotation\; encoding="application/x-tex">f</annotation></semantics>$*, along with a witness $<semantics>\mathrm{trans}(f,u)<annotation\; encoding="application/x-tex">trans(f,u)</annotation></semantics>$. As with Segal types, this single contractibility condition suffices to ensure that this action is coherently functorial. It also ensures that the fibers $<semantics>C(x)<annotation\; encoding="application/x-tex">C(x)</annotation></semantics>$ are discrete, and that the total space $<semantics>{\sum}_{x:A}C(x)<annotation\; encoding="application/x-tex">\backslash sum\_\{x:A\}\; C(x)</annotation></semantics>$ is Segal.

In particular, for any Segal type $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ and any $<semantics>a:A<annotation\; encoding="application/x-tex">a:A</annotation></semantics>$, the hom-functor $<semantics>{\mathrm{hom}}_{A}(a,-):A\to \mathcal{U}<annotation\; encoding="application/x-tex">\backslash hom\_A(a,-)\; :A\; \backslash to\; \backslash mathcal\{U\}</annotation></semantics>$ is covariant. The Yoneda lemma says that for any covariant $<semantics>C:A\to \mathcal{U}<annotation\; encoding="application/x-tex">C:A\backslash to\; \backslash mathcal\{U\}</annotation></semantics>$, evaluation at $<semantics>(a,{\mathrm{id}}_{a})<annotation\; encoding="application/x-tex">(a,id\_a)</annotation></semantics>$ defines an equivalence

$$<semantics>(\prod _{x:A}{\mathrm{hom}}_{A}(a,x)\to C(x))\simeq C(a)<annotation\; encoding="application/x-tex">\; \backslash Big(\backslash prod\_\{x:A\}\; \backslash hom\_A(a,x)\; \backslash to\; C(x)\backslash Big)\; \backslash simeq\; C(a)\; </annotation></semantics>$$

The usual proof of the Yoneda lemma applies, except that it’s simpler since we don’t need to check naturality or functoriality; in the “synthetic” world all of that comes for free.

More generally, we have a *dependent Yoneda lemma*, which says that for any covariant $<semantics>C:({\sum}_{x:A}{\mathrm{hom}}_{A}(a,x))\to \mathcal{U}<annotation\; encoding="application/x-tex">C\; :\; \backslash Big(\backslash sum\_\{x:A\}\; \backslash hom\_A(a,x)\backslash Big)\; \backslash to\; \backslash mathcal\{U\}</annotation></semantics>$, we have a similar equivalence

$$<semantics>(\prod _{x:A}\prod _{f:{\mathrm{hom}}_{A}(a,x)}C(x,f))\simeq C(a,{\mathrm{id}}_{a}).<annotation\; encoding="application/x-tex">\; \backslash Big(\backslash prod\_\{x:A\}\; \backslash prod\_\{f:\backslash hom\_A(a,x)\}\; C(x,f)\backslash Big)\; \backslash simeq\; C(a,id\_a).\; </annotation></semantics>$$

This should be compared with the universal property of identity-elimination (path induction) in ordinary homotopy type theory, which says that for *any* type family $<semantics>C:({\sum}_{x:A}(a=x))\to \mathcal{U}<annotation\; encoding="application/x-tex">C\; :\; \backslash Big(\backslash sum\_\{x:A\}\; (a=x)\backslash Big)\; \backslash to\; \backslash mathcal\{U\}</annotation></semantics>$, evaluation at $<semantics>(a,{\mathrm{refl}}_{a})<annotation\; encoding="application/x-tex">(a,refl\_a)</annotation></semantics>$ defines an equivalence

$$<semantics>(\prod _{x:A}\prod _{f:a=x}C(x,f))\simeq C(a,{\mathrm{refl}}_{a}).<annotation\; encoding="application/x-tex">\; \backslash Big(\backslash prod\_\{x:A\}\; \backslash prod\_\{f:a=x\}\; C(x,f)\backslash Big)\; \backslash simeq\; C(a,refl\_a).\; </annotation></semantics>$$

In other words, the dependent Yoneda lemma really is a “directed” generalization of path induction.

## Rezk types

When is an arrow $<semantics>f:{\mathrm{hom}}_{A}(x,y)<annotation\; encoding="application/x-tex">f\; :\; hom\_A(x,y)</annotation></semantics>$ in a Segal type an isomorphism? Classically, $<semantics>f<annotation\; encoding="application/x-tex">f</annotation></semantics>$ is an isomorphism just when it has a two-sided inverse, but in homotopy type theory more care is needed, for the same reason that we have to be careful when defining what it means for a function to be an equivalence: we want the notion of “being an isomorphism” to be a mere proposition. We could use analogues of any of the equivalent notions of equivalence in Chapter 4 of the HoTT Book, but the simplest is the following:

$$<semantics>\mathrm{isiso}(f):=(\sum _{g:{\mathrm{hom}}_{A}(y,x)}g\circ f={\mathrm{id}}_{x})\times (\sum _{h:{\mathrm{hom}}_{A}(y,x)}f\circ h={\mathrm{id}}_{y})<annotation\; encoding="application/x-tex">\; isiso(f)\; :=\; \backslash left(\backslash sum\_\{g\; :\; hom\_A(y,x)\}\; g\; \backslash circ\; f\; =\; id\_x\backslash right)\; \backslash times\; \backslash left(\backslash sum\_\{h\; :\; hom\_A(y,x)\}\; f\; \backslash circ\; h\; =\; id\_y\; \backslash right)\; </annotation></semantics>$$

An element of this type consists of a left inverse and a right inverse together with witnesses that the respective composites with $<semantics>f<annotation\; encoding="application/x-tex">f</annotation></semantics>$ define identities. It is easy to prove that $<semantics>g=h<annotation\; encoding="application/x-tex">g\; =\; h</annotation></semantics>$, so that $<semantics>f<annotation\; encoding="application/x-tex">f</annotation></semantics>$ is an isomorphism if and only if it admits a two-sided inverse, but the point is that any pair of terms in the type $<semantics>\mathrm{isiso}(f)<annotation\; encoding="application/x-tex">isiso(f)</annotation></semantics>$ are equal (i.e., $<semantics>\mathrm{isiso}(f)<annotation\; encoding="application/x-tex">isiso(f)</annotation></semantics>$ is a mere proposition), which would not be the case for the more naive definition.

The type of isomorphisms from $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ to $<semantics>y<annotation\; encoding="application/x-tex">y</annotation></semantics>$ in $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is then defined to be

$$<semantics>(x{\cong}_{A}y):=\sum _{f:{\mathrm{hom}}_{A}(x,y)}\mathrm{isiso}(f).<annotation\; encoding="application/x-tex">\; (x\; \backslash cong\_A\; y)\; :=\; \backslash sum\_\{f\; :\; \backslash hom\_A(x,y)\}\; isiso(f).\; </annotation></semantics>$$

Identity arrows are in particular isomorphisms, so by identity-elimination there is a map

$$<semantics>\prod _{x,y:A}(x{=}_{A}y)\to (x{\cong}_{A}y)<annotation\; encoding="application/x-tex">\; \backslash prod\_\{x,y:\; A\}\; (x\; =\_A\; y)\; \backslash to\; (x\; \backslash cong\_A\; y)\; </annotation></semantics>$$

and we say that a Segal type $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is **Rezk complete** if this map is an equivalence, in which case $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is a **Rezk type**.

## Coherent adjunctions

Similarly, it is somewhat delicate to define homotopy correct types of adjunction data that are contractible when they are inhabited. In the final section to our paper, we compare *transposing adjunctions*, by which we mean functors $<semantics>f:A\to B<annotation\; encoding="application/x-tex">f\; :\; A\; \backslash to\; B</annotation></semantics>$ and $<semantics>u:B\to A<annotation\; encoding="application/x-tex">u\; :\; B\; \backslash to\; A</annotation></semantics>$ (i.e. functions between Segal types) together with a fiberwise equivalence

$$<semantics>\prod _{a:A,b:B}{\mathrm{hom}}_{A}(a,ub)\simeq {\mathrm{hom}}_{B}(fa,b)<annotation\; encoding="application/x-tex">\; \backslash prod\_\{a\; :A,\; b:\; B\}\; \backslash hom\_A(a,u\; b)\; \backslash simeq\; \backslash hom\_B(f\; a,b)\; </annotation></semantics>$$

with various notions of *diagrammatic adjunctions*, specified in terms of units and counits and higher coherence data.

The simplest of these, which we refer to as a **quasi-diagrammatic adjunction** is specified by a pair of functors as above, natural transformations $<semantics>\eta :{\mathrm{Id}}_{A}\to uf<annotation\; encoding="application/x-tex">\backslash eta:\; Id\_A\; \backslash to\; u\; f</annotation></semantics>$ and $<semantics>\u03f5:fu\to {\mathrm{Id}}_{B}<annotation\; encoding="application/x-tex">\backslash epsilon\; :\; f\; u\; \backslash to\; Id\_B</annotation></semantics>$ (a “natural transformation” is just an arrow in a function-type between Segal types), and witnesses $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$ and $<semantics>\beta <annotation\; encoding="application/x-tex">\backslash beta</annotation></semantics>$ to both of the triangle identities. The incoherence of this type of data has been observed in bicategory theory (it is not cofibrant as a 2-category) and in $<semantics>(\mathrm{\infty},1)<annotation\; encoding="application/x-tex">(\backslash infty,1)</annotation></semantics>$-catgory theory (as a subcomputad of the free homotopy coherent adjunction it is not *parental*). One
homotopically correct type of adjunction data is a **half-adjoint diagrammatic adjunction**, which has additionally a witness that $<semantics>f\alpha :\u03f5\circ fu\u03f5\circ f\eta u\to \u03f5<annotation\; encoding="application/x-tex">f\; \backslash alpha\; :\; \backslash epsilon\; \backslash circ\; f\; u\backslash epsilon\; \backslash circ\; f\backslash eta\; u\; \backslash to\; \backslash epsilon</annotation></semantics>$ and $<semantics>\beta u:\u03f5\circ \u03f5fu\circ f\eta u<annotation\; encoding="application/x-tex">\backslash beta\; u:\; \backslash epsilon\; \backslash circ\; \backslash epsilon\; f\; u\; \backslash circ\; f\; \backslash eta\; u</annotation></semantics>$ commute with the naturality isomorphism for $<semantics>\u03f5<annotation\; encoding="application/x-tex">\backslash epsilon</annotation></semantics>$.

We prove that given Segal types $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ and $<semantics>B<annotation\; encoding="application/x-tex">B</annotation></semantics>$ and functors $<semantics>f:A\to B<annotation\; encoding="application/x-tex">f\; :\; A\; \backslash to\; B</annotation></semantics>$ and $<semantics>u:B\to A<annotation\; encoding="application/x-tex">u\; :\; B\; \backslash to\; A</annotation></semantics>$, the type of half-adjoint diagrammatic adjunctions between them is equivalent to the type of transposing adjunctions. More precisely, if in the notion of transposing adjunction we interpret “equivalence” as a “half-adjoint equivalence”, i.e. a pair of maps $<semantics>\varphi <annotation\; encoding="application/x-tex">\backslash phi</annotation></semantics>$ and $<semantics>\psi <annotation\; encoding="application/x-tex">\backslash psi</annotation></semantics>$ with homotopies $<semantics>\varphi \psi =1<annotation\; encoding="application/x-tex">\backslash phi\; \backslash psi\; =\; 1</annotation></semantics>$ and $<semantics>\psi \varphi =1<annotation\; encoding="application/x-tex">\backslash psi\; \backslash phi\; =\; 1</annotation></semantics>$ and a witness to *one* of the triangle identities for an adjoint equivalence (this is another of the coherent notions of equivalence from the HoTT Book), then these data correspond exactly under the Yoneda lemma to those of a half-adjoint diagrammatic adjunction.

This suggests that similar correspondences for other kinds of coherent equivalences. For instance, if we interpret transposing adjunctions using the “bi-invertibility” notion of coherent equivalence (specification of a separate left and right inverse, as we used above to define isomorphisms in a Segal type), we obtain upon Yoneda-fication a new notion of coherent diagrammatic adjunction, consisting of a unit $<semantics>\eta <annotation\; encoding="application/x-tex">\backslash eta</annotation></semantics>$ and *two* counits $<semantics>\u03f5,\u03f5\prime <annotation\; encoding="application/x-tex">\backslash epsilon,\backslash epsilon\text{\'}</annotation></semantics>$, together with witnesses that $<semantics>\eta ,\u03f5<annotation\; encoding="application/x-tex">\backslash eta,\backslash epsilon</annotation></semantics>$ satisfy one triangle identity and $<semantics>\eta ,\u03f5\prime <annotation\; encoding="application/x-tex">\backslash eta,\backslash epsilon\text{\'}</annotation></semantics>$ satisfy the other triangle identity.

Finally, if the types $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ and $<semantics>B<annotation\; encoding="application/x-tex">B</annotation></semantics>$ are not just Segal but Rezk, we can show that adjoints are literally unique, not just “unique up to isomorphism”. That is, given a functor $<semantics>u:B\to A<annotation\; encoding="application/x-tex">u:B\backslash to\; A</annotation></semantics>$ between Rezk types, the “type of left adjoints to $<semantics>u<annotation\; encoding="application/x-tex">u</annotation></semantics>$” is a mere proposition.

## May 24, 2017

### Christian P. Robert - xi'an's og

**A**n email I got today from Heng Zhou wondered about the validity of the above form of the ARS algorithm. As printed in our book Monte Carlo Statistical Methods. The worry is that in the original version of the algorithm the envelope of the log-concave target f(.) is only updated for rejected values. My reply to the question is that there is no difference in the versions towards returning a value simulated from f, since changing the envelope between simulations does not modify the accept-reject nature of the algorithm. There is no issue of dependence between the simulations of this adaptive accept-reject method, all simulations remain independent. The question is rather one about efficiency, namely does it pay to update the envelope(s) when accepting a new value and I think it does because the costly part is the computation of f(x), rather than the call to the piecewise-exponential envelope. Correct me if I am wrong!

Filed under: Books, Kids, Statistics, University life Tagged: accept-reject algorithm, ARS, log-concave functions, Monte Carlo Statistical Methods, typos, Wally Gilks

### ZapperZ - Physics and Physicists

Over the weekend, cosmologist and author Sean Carroll tweeted about what physics majors should know, namely that "the Standard Model is an SU(3)xSU(2)xU(1) gauge theory, and know informally what that means." My immediate reaction to this was pretty much in line with Brian Skinner's, namely that this is an awfully specific and advanced bit of material to be a key component of undergraduate physics education. (I'm assuming an undergrad context here, because you wouldn't usually talk about a "major" at the high school or graduate school levels.)

I categorize the tweet by Carroll as silly because he has no evidence to back up WHY this is such an important piece of information and knowledge for EVERY physics major. I hate to make my own silly generalization, but I'm going to here. This type of assertion sounds like it is a typical comment made by a theorist working on an esoteric subject matter. There! I've said it, and I'm sure I've offended many people already!

I would like to make another assertion, which is that there are PLENTY (even majority?) of physics majors who got their undergraduate degree without "informally" knowing the meaning of "...

*the Standard Model is an SU(3)xSU(2)xU(1) gauge theory*...", AND..... go on to have a meaningful career in physics. Anyone care to dispute me on that?

If that is true, then Carroll's assertion is meaningless, because there appears to be NO valid reason for why a physics major needs to know that. He/she needs to know QM, CM, and E&M. That much I will give. Orzel even listed these and other subject areas that a typical undergraduate in physics is assumed to know. But a gauge symmetry in the Standard Model? Is this even in the Physics GRE?

Considering that about HALF of B.Sc degree recipients in physics do not go on to graduate school, I can think of many other, MORE IMPORTANT skills and knowledge that we should equipped physics majors. We are trying to make physics majors more "employable" in the marketplace, especially in the private sector. Comments by Carroll simply re-enforced the DISCONNECT that many physics departments have in how they train and educate their students without paying attention to their employment possibilities beyond research and academia. This is highly irresponsible!

I'm glad that Orzel took this head on, because Sean Carroll should know better... or maybe he doesn't, and that's the problem!

Zz.

### Lubos Motl - string vacua and pheno

*This stack of cards may actually be seen in the lower right corner of all graphs produced by GAMBIT. ;-)*

Click at the hyperlink to learn about their project. I have always called for the creation of such systems and it's great that one of them seems to be born by now.

Much of the work of model builders is really about some routine work – one works with some new fields and interaction terms in the Lagrangian, some methods to calculate particle physics predictions, scan the parameter spaces, compute probability distributions, and compare predictions with the experimental data etc.

A key word is "routine": Quantum field theory and its application is nontrivial and one needs to learn many prerequisites before she gets at this level. On the other hand, it's a finite amount of knowledge and the technology has almost always the same character, independently of the particular model beyond the Standard Model that one proposes.

So this collaboration of 30 model builders proposes their code to systematize much of the work. With this help of the computer, lots of human work should be saved and the work should become faster and more effective. Many smart brains could be saved for some more creative work, especially some serious thinking about string theory. We sometimes talk about occupations that may be replaced with robots in a decade – most model builders may be among them.

I believe that experimental teams such as those at the LHC should join and/or develop their own programs that may basically produce the usual ATLAS/CMS papers about all the channels parameterized by a particular theory as outcomes generated by the same program run with different arguments or parameters. Don't the authors of the hundreds of ATLAS/CMS papers feel that they're doing a boring work that would be better done by a computer?

At any rate, most of the today's new hep-ph papers are all about GAMBIT. It's the nine papers [1], [5-7], [9-13].

You may download all the codes. It's weird that most of the archives are either 43 or 44 or 45 megabytes in size although they seem to have a different content. The code is supposed to run on supercomputers such as Prometheus but I think that Promotheus shouldn't be "absolutely required". In March, Physics World published a story about GAMBIT.

Under this avalanche of papers, it's easy to overlook a new paper by Nanopoulos, Li, Maxin who still seem excited about the \({\mathcal F}\)-\(SU(5)\) models even though they had to raise the gluino mass to \(1.9\)-\(2.3\TeV\).

by Luboš Motl (noreply@blogger.com) at May 24, 2017 07:43 AM

## May 23, 2017

### Andrew Jaffe - Leaves on the Line

This week, the New York Times, The Wall Street Journal and Twitter, along with several other news organizations, have all announced that they were attacked by (most likely) Chinese hackers.

I am not quite happy to join their ranks: for the last few months, the traffic on this blog has been vastly dominated by attempts to get into the various back-end scripts that run this site, either by direct password hacks or just denial-of-service attacks. In fact, I only noticed it because the hackers exceeded my bandwidth allowance by a factor of a few (and costing me a few hundred bucks in over-usage charged by my host in the process, unfortunately).

I’ve since attempted to block the attacks by denying access to the IP addresses which have been the most active (mostly from domains that look like 163data.com.cn, for what it’s worth). So, my apologies if any of this results in any problems for anyone else trying to access the blog.

### Andrew Jaffe - Leaves on the Line

More technical stuff, but I’m trying to re-train myself to actually write on this blog, so here goes…

For no good reason other than it was easy, I have added a JSONfeed to this blog. It can be found at http://andrewjaffe.net/blog/feed.json, and accessed from the bottom of the right-hand sidebar if you’re actually reading this at andrewjaffe.net.

What does this mean? JSONfeed is an idea for a sort-of successor to something called RSS, which may stand for really simple syndication, a format for encapsulating the contents of a blog like this one so it can be indexed, consumed, and read in a variety of ways without explicitly going to my web page. RSS was created by developer, writer, and all around web-and-software guru Dave Winer, who also arguably invented — and was certainly part of the creation of — blogs and podcasting. Five or ten years ago, so-called RSS readers were starting to become a common way to consume news online. NetNewsWire was my old favourite on the Mac, although its original versions by Brent Simmons were much better than the current incarnation by a different software company; I now use something called Reeder. But the most famous one was Google Reader, which Google discontinued in 2013, thereby killing off most of the RSS-reader ecosystem.

But RSS is not dead: RSS readers still exist, and it is still used to store and transfer information between web pages. Perhaps most importantly, it is the format behind subscriptions to podcasts, whether you get them through Apple or Android or almost anyone else.

But RSS is kind of clunky, because it’s built on something called XML, an ugly but readable format for structuring information in files (HTML, used for the web, with all of its < and > “tags”, is a close cousin). Nowadays, people use a simpler family of formats called JSON for many of the same purposes as XML, but it is quite a bit easier for humans to read and write, and (not coincidentally) quite a bit easier to create computer programs to read and write.

So, finally, two more web-and-software developers/gurus, Brent Simmons and Manton Reece realised they could use JSON for the same purposes as RSS. Simmons is behind NewNewsWire and Reece’s most recent project is an “indie microblogging” platform (think Twitter without the giant company behind it), so they both have an interest in these things. And because JSON is so comparatively easy to use, there is already code that I could easily add to this blog so it would have its own JSONfeed. So I did it.

So it’s easy to create a JSONfeed. What there isn’t — so far — are any newsreaders like NetNewsWire or Reeder that can ingest them. (In fact, Maxime Vaillancourt apparently wrote a web-based reader in about an hour, but it may already be overloaded…). Still, looking forward to seeing what happens.

### Symmetrybreaking - Fermilab/SLAC

Protons are colliding once again in the Large Hadron Collider.

This morning at CERN, operators nudged two high-energy beams of protons into a collision course inside the world’s largest and most energetic particle accelerator, the Large Hadron Collider. These first stable beams inside the LHC since the extended winter shutdown usher in another season of particle hunting.

The LHC’s 2017 run is scheduled to last until December 10. The improvements made during the winter break will ensure that scientists can continue to search for new physics and study rare subatomic phenomena. The machine exploits Albert Einstein’s principle that energy and matter are equivalent and enables physicists to transform ordinary protons into the rare massive particles that existed when our universe was still in its infancy.

“Every time the protons collide, it’s like panning for gold,” says Richard Ruiz, a theorist at Durham University. “That’s why we need so much data. It’s very rare that the LHC produces something interesting like a Higgs boson, the subatomic equivalent of a huge gold nugget. We need to find lots of these rare particles so that we can measure their properties and be confident in our results.”

During the LHC’s four-month winter shutdown, engineers replaced one of its main dipole magnets and carried out essential upgrades and maintenance work. Meanwhile, the LHC experiments installed new hardware and revamped their detectors. Over the last several weeks, scientists and engineers have been performing the final checks and preparations for the first “stable beams” collisions.

“There’s no switch for the LHC that instantly turns it on,” says Guy Crockford, an LHC operator. “It’s a long process, and even if it’s all working perfectly, we still need to check and calibrate everything. There’s a lot of power stored in the beam and it can easily damage the machine if we’re not careful.”

In preparation for data-taking, the LHC operations team first did a cold checkout of the circuits and systems without beam and then performed a series of dress rehearsals with only a handful of protons racing around the machine.

“We set up the machine with low intensity beams that are safe enough that we could relax the safety interlocks and make all the necessary tweaks and adjustments,” Crockford says. “We then deliberately made the proton beams unstable to check that all the loose particles were caught cleanly. It’s a long and painstaking process, but we need complete confidence in our settings before ramping up the beam intensity to levels that could easily do damage to the machine.”

The LHC started collisions for physics with only three proton bunches per beam. Over the course of the next month, the operations team will gradually increase the number of proton bunches until they have 2760 per beam. The higher proton intensity greatly increases the rate of collisions, enabling the experiments to collect valuable data at a much faster rate.

“We’re always trying to improve the machine and increase the number of collisions we deliver to the experiments,” Crockford says. “It’s a personal challenge to do a little better every year.”

### Emily Lakdawalla - The Planetary Society Blog

### Andrew Jaffe - Leaves on the Line

It’s not that often that I can find a reason to write about both astrophysics and music — my obsessions, vocations and avocations — at the same time. But the recent release of Scott Walker’s (certainly weird, possibly wonderful) new record Bish Bosch has given me just such an excuse: Track 4 is a 21-minute opus of sorts, entitled “SDSS1416+13B (Zercon, A Flagpole Sitter)”. The title seems a random collection of letters, numbers and words, but that’s not what it is: SDSS1416+13B is the (very slightly mangled) identification of an object in the Sloan Digital Sky Survey (SDSS) catalog — 1416+13B means that it is located at Right Ascension 14^{h}16^{m} and Declination 13° (actually, its full name is SDSS J141624.08+134826.7 which gives the location more precisely) and “B” denotes that it’s actually the second of two objects (the other one is unsurprisingly called “A”).

In fact it’s a pretty interesting object: it was actually discovered not by SDSS alone, but by cross-matching with another survey, the UK Infrared Deep Sky Survey (UKIDSS) and looking at the images by eye. It turns out that the two components are a binary system made up of two brown dwarfs — objects that aren’t massive enough to burn hydrogen via nuclear fusion, but are more massive than even the heaviest planets, often big enough to form at the centre of their own stellar systems, and heavy enough have some nuclear reactions in their core. In fact, the UKIDSS survey has been one of the best ways to find such comparatively cool objects; my colleagues Daniel Mortlock and Steve Warren found one of the coolest known brown dwarfs in UKIDSS in 2007, using techniques very similar to those they also used to find the most distant quasar yet known, recounted by Daniel in a guest-post here. Like that object, SDSS1416+13B is one of the coolest such objects ever found.

What does all this have to do with Scott Walker? I have no idea. Since he started singing as a member of the Walker Brothers in the 60s — and even more so since his 70s solo records, Walker has been known for his classical-sounding baritone, though with his mannered, massive vibrato, he always sounds a bit like a rocker’s caricature of a classical singer. I’ve always thought it was more force of personality than actual skill that drew people — especially here in the UK — to him.

His latest, Bish Bosch, the third in a purported trilogy of records he’s made since resurfacing in the mid-1990s, veers between mannered art-songs and rock’n’roll, silences punctuated with electric guitars, fart-sounds and trumpets.

The song “SDSS1416” itself is an (I assume intentionally funny?) screed, alternating sophomoric insults (my favourite is “don’t go to a mind reader, go to a palmist; I know you’ve got a palm”) with recitations of Roman numerals and, finally, the only link to observations of a brown dwarf I can find, “Infrared, infrared/ I could drop/ into the darkness.” Your guess is as good as mine. It’s compelling, but I can’t tell if that’s as an epic or a train wreck.

### Andrew Jaffe - Leaves on the Line

In further pop-culture crossover news, I was pleased to see this paragraph in John Keane’s review of Alan Ryan’s “On Politics” in this weekend’s Financial Times:

Ryan sees this period [the 1940s] as the point of triumph of liberal democracy against its Fascist and Stalinist opponents. Closer attention shows this decade was instead a moment of what physicists call dark energy: the universe of meaning of democracy underwent a dramatic expansion, in defiance of the cosmic gravity of contemporary events. The ideal of monitory democracy was born.

Not a bad metaphor. Nice to see that the author, a professor of Politics from Sydney, is paying attention to the stuff that really matters.

### Andrew Jaffe - Leaves on the Line

A week ago, I finished my first time teaching our second-year course in quantum mechanics. After a bit of a taster in the first year, the class concentrates on the famous Schrödinger equation, which describes the properties of a particle under the influence of an external force. The simplest version of the equation is just This relates the so-called wave function, ψ, to what we know about the external forces governing its motion, encoded in the Hamiltonian operator, Ĥ. The wave function gives the probability (technically, the probability amplitude) for getting a particular result for any measurement: its position, its velocity, its energy, etc. (See also this excellent public work by our department’s artist-in-residence.)

Over the course of the term, the class builds up the machinery to predict the properties of the hydrogen atom, which is the canonical real-world system for which we need quantum mechanics to make predictions. This is certainly a sensible endpoint for the 30 lectures.

But it did somehow seem like a very old-fashioned way to teach the course. Even back in the 1980s when I first took a university quantum mechanics class, we learned things in a way more closely related to the way quantum mechanics is used by practicing physicists: the mathematical details of Hilbert spaces, path integrals, and Dirac Notation.

Today, an up-to-date quantum course would likely start from the perspective of quantum information, distilling quantum mechanics down to its simplest constituents: qbits, systems with just two possible states (instead of the infinite possibilities usually described by the wave function). The interactions become less important, superseded by the information carried by those states.

Really, it should be thought of as a full year-long course, and indeed much of the good stuff comes in the second term when the students take “Applications of Quantum Mechanics” in which they study those atoms in greater depth, learn about fermions and bosons and ultimately understand the structure of the periodic table of elements. Later on, they can take courses in the mathematical foundations of quantum mechanics, and, yes, on quantum information, quantum field theory and on the application of quantum physics to much bigger objects in “solid-state physics”.

Despite these structural questions, I was pretty pleased with the course overall: the entire two-hundred-plus students take it at the beginning of their second year, thirty lectures, ten ungraded problem sheets and seven in-class problems called “classworks”. Still to come: a short test right after New Year’s and the final exam in June. Because it was my first time giving these lectures, and because it’s such an integral part of our teaching, I stuck to to the same notes and problems as my recent predecessors (so many, many thanks to my colleagues Paul Dauncey and Danny Segal).

Once the students got over my funny foreign accent, bad board handwriting, and worse jokes, I think I was able to get across both the mathematics, the physical principles and, eventually, the underlying weirdness, of quantum physics. I kept to the standard Copenhagen Interpretation of quantum physics, in which we think of the aforementioned wavefunction as a real, physical thing, which evolves under that Schrödinger equation — except when we decide to make a measurement, at which point it undergoes what we call collapse, randomly and seemingly against causality: this was Einstein’s “spooky action at a distance” which seemed to indicate nature playing dice with our Universe, in contrast to the purely deterministic physics of Newton and Einstein’s own relativity. No one is satisfied with Copenhagen, although a more coherent replacement has yet to be found (I won’t enumerate the possibilities here, except to say that I find the proliferating multiverse of Everett’s Many-Worlds interpretation ontologically extravagant, and Chris Fuchs’ Quantum Bayesianism compelling but incomplete).

I am looking forward to getting this year’s SOLE results to find out for sure, but I think the students learned something, or at least enjoyed trying to, although the applause at the end of each lecture seemed somewhat tinged with British irony.

### Tommaso Dorigo - Scientificblogging

This is the first of a series of posts that will publish the results of artistic work by high-school students of three schools in Venice, who participate in a contest and exposition connected to the initiative "Art and Science across Italy", an initiative of the network CREATIONS, funded by the Horizon 2020 programme

## May 22, 2017

### CERN Bulletin

On Thursday June 1^{st} at 12.15, Fabiola Gianotti, our Director-General, will fire the starting shot for the 47^{th} Relay Race.

This Race is above all a festive CERN event, open for runners and walkers, as well as the people cheering them on throughout the race, and those who wish to participate in the various activities organised between 11.30 and 14.30 out on the lawn in front of Restaurant 1.

In order to make this sports event accessible for everyone, our Director-General will allow for flexible lunch hours on the day, applicable for all the members of personnel.

An alert for the closure of roads will be send out on the day of the event.

The Staff Association and the CERN Running Club thank you in advance for your participation and your continued support throughout the years.

This year the CERN Running Club has announced the participation of locally and internationally renowned runners, no less!

A bit over a week from the Relay Race of 1^{st} June, the number of teams is going up nicely (already almost 40).

Among them, we will have three teams this year from our main partner Berthie Sport, and I can tell you that they are not coming just for fun!

The ladies’ team has been built from the best runners in the Pays de Gex, including Laetitia Matlet, winner of the Challenge des courses à pied du Pays de Gex, and Isabelle Marchand, winner of the Foulées de Crozet.

But the most impressive will definitely be the men’s team, with the presence of several top-level runners:

- Tristan Le Lay, triathlete of European level, 4:15 on half ironman
- Pierre Baque, winner of the SaintéLyon Relay, 1:10 on half marathon
- Ludovic Pommeret, winner of UTMB, one of the top ultra-trail runners in the world!

You can start placing your bets on the new race record! :)

### CERN Bulletin

# Small Capella

## Friday 2 June at 18.00

CERN Meyrin, Main Auditorium

**Free admission**

Moscow chamber choir Small Capella arose within the walls of Children‘s musical school No. 10, and evolved over the years into a mixed choir of people of various age and occupation, open to anyone fond of choral music.

The repertoire includes Russian and foreign classical music, sacred music, folk songs, contemporary choral compositions.

The concert will include solo vocal and piano pieces.

### CERN Bulletin

**Summer is coming, enjoy our offers for the aquatic parcs!**

**Walibi **:

__Tickets "Zone terrestre"__: **24 €** instead of 30 €.

__Access to Aqualibi__: **5 €** instead of 6 € on presentation of your SA member ticket.

*Free for children under 100 cm.*

Car park free.

* * * * *

**Aquaparc **:

__Day ticket__:

– __Children__: **33 CH**F instead of 39 CHF

– __Adults __: **33 CHF** instead of 49 CHF

*Bonus! Free for children under 5.*

### CERN Bulletin

In the first semester of each year, the Staff Association (SA) invites its members to attend and participate in the Ordinary General Assembly (OGA).

This year the OGA will be held on __Thursday, 29 June 2017 from 15.30 to 17.30__, Main Auditorium, Meyrin (500-1-001).

During the Ordinary General Assembly, the activity and financial reports of the SA are presented and submitted for approval to the members. This is the occasion to get a global view on the activities of the SA, its management, and an opportunity to express your opinion, particularly by taking part in votes. Other items are listed on the agenda, as proposed by the Staff Council.

# Who can vote?

Ordinary members (MPE) of the SA can take part in all votes. Associated members (MPA) of the SA and/or affiliated pensioners have a right to vote on those topics that are of direct interest to them.

# Who can give their opinion, and how?

The Ordinary General Assembly is also the opportunity for members of the SA to express themselves through the addition of discussion points to the agenda. For these points to be subjected to a vote, the request must be introduced in writing to the President of the Staff Association, at least 20 days before the General Assembly, and by at least 20 members of the SA. Additionally, members of the SA can ask the OGA to have a discussion on a specific point, after expiration of the agenda, but no decision shall be taken based on these discussions.

# Can we contest the decisions?

Any decision taken by the Ordinary General Assembly can be contested through a referendum as defined in the Statute of the Staff Association.

**Do not hesitate, take part in your Ordinary General Assembly on 29 June 2017. Come and make your voice count, and seize this occasion to exchange with your staff delegates!**

### CERN Bulletin

Le GAC organise des permanences avec entretiens individuels qui se tiennent le dernier mardi de chaque mois, __sauf en juin, juillet et décembre__.

La prochaine permanence se tiendra le :

**Mardi 30 mai de 13 h 30 à 16 h 00**

**Salle de réunion de l’Association du personnel**

Les permanences du Groupement des Anciens sont ouvertes aux bénéficiaires de la Caisse de pensions (y compris les conjoints survivants) et à tous ceux qui approchent de la retraite.

Nous invitons vivement ces derniers à s’associer à notre groupement en se procurant, auprès de l’Association du personnel, les documents nécessaires.

Informations : http://gac-epa.org/

Formulaire de contact : http://gac-epa.org/Organization/ContactForm/ContactForm-fr.php

## May 21, 2017

### Tommaso Dorigo - Scientificblogging

## May 20, 2017

### Geraint Lewis - Cosmic Horizons

But I did get a chance to do some recreational mathematics, spurred on my a story in the news. It's to do with a problem presented at the 2017 Raytheon MATHCOUNTS® National Competition and reported in the New York Times. Here's the question as presented in the press:

Kudos to 13 year old Texan, Luke Robitialle, who got this right.

With a little thought, you should be able to realise that the answer is 25. For any particular chick, there are four potential out comes, each with equal probability. Either the chick is

- pecked from the left
- pecked from the right
- pecked from left and right
- not pecked at all

*distribution*of unpecked chicks? What I mean by this is that they peck left or right at random, there might be 24 unpecked chicks for one group of a hundred chicks, 25 for the next, and 23 for the next. So, the question is, given a large number of 100 chick experiments, what's the distribution of unpecked chicks?

^{100}-1, and represent it as a binary number, then that will be a random sampling of the pecking order (pecking order, get it!) As an example, all chicks peck to the left would be 100 0s in binary, where as all the chicks peck to the right would be 100 1s in binary.

Let's try a randomly drawn integer in the range. We get (in base 10) 333483444300232384702347234. In binary this is

`0000000000010001001111011001110111011011110101101010111100001100011100100110100100000111011111100010`

`So, the first bunch of chicks peck to the left, then we have a mix of right to left pecks. `

`But how many of these chicks are unpecked (remembering what the original question)? Well, for any particular chick, it will be unpecked if the chick to its left pecks to the left, and the chick to its right pecks to the right. So, we're looking for sequences of '001' and '011', with the middle digit representing the chick we are interested in. `

`So, we can chick this into a little python code (had to learn it, all the cool kids are using it these days) and this is what I have`

`There is a little extra in there to account for the fact that the chicks are sitting in a circle, but as you can see, the code is quite compact.`

`OK. Let's run for the 100 chicks in the question. What do we get?`

`Yay! The unpecks peak at 25, but there is a nice distribution (which, I am sure, must have an analytic solution somewhere. `

`But given the simplicity of the code, I can easily change the number of chicks. What about 10 chicks in circle?`

`Hmmm. Interesting. What about 1000 chicks?`

`And 33 chicks?`

`Most likely number of unpecked chicks is 8, but again, a nice distribution. `

`Now, you might be sitting there wondering why the heck I am doing this? Well, firstly, because it is fun! And interesting! It's a question and it is fun to find the answer. `

`Secondly it is not obvious what the distribution would be, and how complex it would be to derive, or even if it exists, and so a numerical approach allows us to find an answer. `

`Finally, I can easily generalize this to questions like "what if the left pecks are more likely than right pecks by a factor of two, what would the distribution be like?" It would just take a couple of lines of code and I would have an answer. `

`And if you can't see how such curiosity led examinations are integral to science, then you don't know what science is.`

## May 19, 2017

### Emily Lakdawalla - The Planetary Society Blog

### Emily Lakdawalla - The Planetary Society Blog

## May 18, 2017

### ZapperZ - Physics and Physicists

First of all, the measure of something to be "easy" or "difficult" it itself is subjective. What is easy to some, can easily be difficult to others (see what I did there?). Meryl Streep can easily memorize pages and pages of dialog, something that I find difficult to do because I am awful at memorization. But yet, I'm sure I can solve many types of differential equations that she finds difficult. So already, there is a degree of "subjectiveness" to this.

But what is more important here is that, in science, for something to be considered as a valid description of something, it must be QUANTIFIABLE. In other words, a number associated with that description can be measured or obtained.

Let's apply this to an example. I can ask: How difficult or easy it is to stop a 100 kg moving mass? So, what am I actually asking here when I ask if it is "easy" or "difficult"? It is vague. However, I can specify that if I use less force to make the object come to a complete stop over a specific distance, then this is EASIER than if I have to use a larger force to do the same thing.

Now THAT is more well-defined, because I am using "easy" or "difficult" as a measure of the amount of force I have to apply. In fact, I can omit the use of the words "easy" and "difficult", and simply ask for the force needed to stop the object. That is a question that is well-defined and quantifiable, such that a quantitative comparison can be made.

Let's come back to the original question that was the impetus of this post. This person asked if it is easier to heat things rather than to cool things. So the question now is, what does it mean for it to be "easy" to heat or cool things. One measure can be that, for a constant heat transfer, how long in time does it take to heat or cool the object by the same change in temperature? So in this case, the measure of time taken to heat and cool the object by the same amount of temperature change is the measure of "easy" or "difficult". One can compare time taken to heat the object by, say, 5 Celsius, versus time taken to cool the object by the same temperature change. Now this, is a more well-defined question.

I bring this up because I often see many ordinary conversation, discussion, news reports, etc.. etc. in which statements and descriptions made appear to be clear and to make sense, when in reality, many of these are really empty statements that are ambiguous, and sometime meaningless. Describing something to be easy or difficult appears to be a "simple" and clear statement or description, but if you think about it carefully, it isn't! Ask yourself if the criteria to classify something to be easy, easier, difficult, more difficult, etc... etc. is plainly evident and universally agreed upon. Did the statement that says "such and such undermines so-and-so" is actually clear on what it is saying? What exactly does "undermines" mean in this case, and what is the measure of it?

Science/Physics education has the ability to impart this kind of analytical skills, and to impart this kind of thinking to the students, especially if they are not specializing in STEM subjects. In science, the nature of the question we ask can often be as important as the answers that we seek. This is because unless we clearly define what it is that we are asking, then we can't know where to look for the answers. This is a lesson that many people in the public need to learn and to be aware of, especially in deciphering many of the things we see in the media right now.

It is why science education is invaluable to everyone.

Zz.

### Clifford V. Johnson - Asymptotia

Well, yesterday evening and today I've got an entirely different hat - SF short story writer! First let me apologize for faking it to all my friends reading who are proper short story writers with membership cards and so on. Let me go on to explain:

I don't think I'm allowed to tell you the full details yet, but the current editor of an annual science fiction anthology got in touch back in February and told me about an idea they wanted to try out. They normally have their usual batch of excellent science fiction stories (from various writers) in the book, ending with a survey of some visual material such as classic SF covers, etc.... but this year they decided to do something different. Instead of the visual survey thing, why not have one of the stories be visual? In other words, a graphic novella (I suppose that's what you'd call it).

After giving them several opportunities to correct their obvious error, which went a bit like this: [...] Click to continue reading this post

The post Writing Hat! appeared first on Asymptotia.

## May 17, 2017

### Tommaso Dorigo - Scientificblogging

*Dr. Alex Durig (see picture) is a professional freelance writer, with a PhD in social psychology from Indiana University (1992). He has authored seven books in his specialization of perception and logic. He claims to have experienced great frustration resolving his experience of perception and logic when it comes to physics, but he says he no longer feels crazy, ever since Anomaly! was published. So I am offering this space to him to hear what he has to say about that...*

*------*

**On Dorigo's**

*Anomaly!*and the Social Psychology of Professional Discourse in Physics, by Alex Durig## May 16, 2017

### Symmetrybreaking - Fermilab/SLAC

At a recent workshop on blind analysis, researchers discussed how to keep their expectations out of their results.

Scientific experiments are designed to determine facts about our world. But in complicated analyses, there’s a risk that researchers will unintentionally skew their results to match what they were expecting to find. To reduce or eliminate this potential bias, scientists apply a method known as “blind analysis.”

Blind studies are probably best known from their use in clinical drug trials, in which patients are kept in the dark about—or blind to—whether they’re receiving an actual drug or a placebo. This approach helps researchers judge whether their results stem from the treatment itself or from the patients’ belief that they are receiving it.

Particle physicists and astrophysicists do blind studies, too. The approach is particularly valuable when scientists search for extremely small effects hidden among background noise that point to the existence of something new, not accounted for in the current model. Examples include the much-publicized discoveries of the Higgs boson by experiments at CERN’s Large Hadron Collider and of gravitational waves by the Advanced LIGO detector.

“Scientific analyses are iterative processes, in which we make a series of small adjustments to theoretical models until the models accurately describe the experimental data,” says Elisabeth Krause, a postdoc at the Kavli Institute for Particle Astrophysics and Cosmology, which is jointly operated by Stanford University and the Department of Energy’s SLAC National Accelerator Laboratory. “At each step of an analysis, there is the danger that prior knowledge guides the way we make adjustments. Blind analyses help us make independent and better decisions.”

Krause was the main organizer of a recent workshop at KIPAC that looked into how blind analyses could be incorporated into next-generation astronomical surveys that aim to determine more precisely than ever what the universe is made of and how its components have driven cosmic evolution.

### Black boxes and salt

One outcome of the workshop was a finding that there is no one-size-fits-all approach, says KIPAC postdoc Kyle Story, one of the event organizers. “Blind analyses need to be designed individually for each experiment.”

The way the blinding is done needs to leave researchers with enough information to allow a meaningful analysis, and it depends on the type of data coming out of a specific experiment.

A common approach is to base the analysis on only some of the data, excluding the part in which an anomaly is thought to be hiding. The excluded data is said to be in a “black box” or “hidden signal box.”

Take the search for the Higgs boson. Using data collected with the Large Hadron Collider until the end of 2011, researchers saw hints of a bump as a potential sign of a new particle with a mass of about 125 gigaelectronvolts. So when they looked at new data, they deliberately quarantined the mass range around this bump and focused on the remaining data instead.

They used that data to make sure they were working with a sufficiently accurate model. Then they “opened the box” and applied that same model to the untouched region. The bump turned out to be the long-sought Higgs particle.

That worked well for the Higgs researchers. However, as scientists involved with the Large Underground Xenon experiment reported at the workshop, the “black box” method of blind analysis can cause problems if the data you’re expressly not looking at contains rare events crucial to figuring out your model in the first place.

LUX has recently completed one of the world’s most sensitive searches for WIMPs—hypothetical particles of dark matter, an invisible form of matter that is five times more prevalent than regular matter. LUX scientists have done a lot of work to guard LUX against background particles—building the detector in a cleanroom, filling it with thoroughly purified liquid, surrounding it with shielding and installing it under a mile of rock. But a few stray particles make it through nonetheless, and the scientists need to look at all of their data to find and eliminate them.

For that reason, LUX researchers chose a different blinding approach for their analyses. Instead of using a “black box,” they use a process called “salting.”

LUX scientists not involved in the most recent LUX analysis added fake events to the data—simulated signals that just look like real ones. Just like the patients in a blind drug trial, the LUX scientists didn’t know whether they were analyzing real or placebo data. Once they completed their analysis, the scientists that did the “salting” revealed which events were false.

A similar technique was used by LIGO scientists, who eventually made the first detection of extremely tiny ripples in space-time called gravitational waves.

### High-stakes astronomical surveys

The Blind Analysis workshop at KIPAC focused on future sky surveys that will make unprecedented measurements of dark energy and the Cosmic Microwave Background—observations that will help cosmologists better understand the evolution of our universe.

Dark energy is thought to be a force that is causing the universe to expand faster and faster as time goes by. The CMB is a faint microwave glow spread out over the entire sky. It is the oldest light in the universe, left over from the time the cosmos was only 380,000 years old.

To shed light on the mysterious properties of dark energy, the Dark Energy Science Collaboration is preparing to use data from the Large Synoptic Survey Telescope, which is under construction in Chile. With its unique 3.2-gigapixel camera, LSST will image billions of galaxies, the distribution of which is thought to be strongly influenced by dark energy.

“Blinding will help us look at the properties of galaxies picked for this analysis independent of the well-known cosmological implications of preceding studies,” DESC member Krause says. One way the collaboration plans on blinding its members to this prior knowledge is to distort the images of galaxies before they enter the analysis pipeline.

Not everyone in the scientific community is convinced that blinding is necessary. Blind analyses are more complicated to design than non-blind analyses and take more time to complete. Some scientists participating in blind analyses inevitably spend time looking at fake data, which can feel like a waste.

Yet others strongly advocate for going blind. KIPAC researcher Aaron Roodman, a particle-physicist-turned-astrophysicist, has been using blinding methods for the past 20 years.

“Blind analyses have already become pretty standard in the particle physics world,” he says. “They’ll be also crucial for taking bias out of next-generation cosmological surveys, particularly when the stakes are high. We’ll only build one LSST, for example, to provide us with unprecedented views of the sky.”

### John Baez - Azimuth

Here you can see the slides of a talk I’m giving:

• The dodecahedron, the icosahedron and E_{8}, Annual General Meeting of the Hong Kong Mathematical Society, Hong Kong University of Science and Technology.

It’ll take place on 10:50 am Saturday May 20th in Lecture Theatre G. You can see the program for the whole meeting here.

The slides are in the form of webpages, and you can see references and some other information tucked away at the bottom of each page.

In preparing this talk I learned more about the geometric McKay correspondence, which is a correspondence between the simply-laced Dynkin diagrams (also known as ADE Dynkin diagrams) and the finite subgroups of

There are different ways to get your hands on this correspondence, but the *geometric* way is to resolve the singularity in where is such a finite subgroup. The variety has a singularity at the origin–or more precisely, the point coming from the origin in To make singularities go away, we ‘resolve’ them. And when you take the ‘minimal resolution’ of this variety (a concept I explain here), you get a smooth variety with a map

which is one-to-one except at the origin. The points that map to the origin lie on a bunch of Riemann spheres. There’s one of these spheres for each dot in some Dynkin diagram—and two of these spheres intersect iff their two dots are connected by an edge!

In particular, if is the double cover of the rotational symmetry group of the dodecahedron, the Dynkin diagram we get this way is :

The basic reason is connected to the icosahedron is that the icosahedral group is generated by rotations of orders 2, 3 and 5 while the Dynkin diagram has ‘legs’ of length 2, 3, and 5 if you count right:

In general, whenever you have a triple of natural numbers obeying

you get a finite subgroup of that contains rotations of orders and a simply-laced Dynkin diagram with legs of length The three most exciting cases are:

• : the tetrahedron, and

• : the octahedron, and

• : the icosahedron, and

But the puzzle is this: why does resolving the singular variety gives a smooth variety with a bunch of copies of the Riemann sphere sitting over the singular point at the origin, with these copies intersecting in a pattern given by a Dynkin diagram?

It turns out the best explanation is in here:

• Klaus Lamotke, *Regular Solids and Isolated Singularities*, Vieweg & Sohn, Braunschweig, 1986.

In a nutshell, you need to start by blowing up at the origin, getting a space containing a copy of on which acts. The space has further singularities coming from the rotations of orders and in . When you resolve these, you get more copies of which intersect in the pattern given by a Dynkin diagram with legs of length and

I would like to understand this better, and more vividly. I want a really clear understanding of the minimal resolution For this I should keep rereading Lamotke’s book, and doing more calculations.

I do, however, have a nice vivid picture of the singular space For that, read my talk! I’m hoping this will lead, someday, to an equally appealing picture of its minimal resolution.

## May 15, 2017

### Clifford V. Johnson - Asymptotia

I've a train to catch and so I did not have time to think of a better title. Sorry. Anyway, for those of you who follow the more technical side of what I do, above is a screen shot to the abstract of a paper to appear tomorrow/today on the arXiv. I'll try to find some time to say more about it, but I can't promise anything since I've got to finish writing another paper today (on the train ride), and then turn myself away from all this for a little while to work on some other things. The abstract should be [...] Click to continue reading this post

The post Bolt those Engines Down… appeared first on Asymptotia.

## May 12, 2017

### The n-Category Cafe

*Guest post by José Siqueira*

We began our journey in the second Kan Extension Seminar with a discussion of the classical concept of * Lawvere theory *, facilitated by Evangelia. Together with the concept of a model, this technology allows one to encapsulate the behaviour of algebraic structures defined by collections of $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$-ary operations subject to axioms (such as the ever-popular groups and rings) in a functorial setting, with the added flexibility of easily transferring such structures to arbitrary underlying categories $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ with finite products (rather than sticking with $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$), naturally leading to important notions such as that of a Lie group.

Throughout the seminar, many features of Lawvere theories and connections to other concepts were unearthed and natural questions were addressed — notably for today’s post, we have established a correspondence between Lawvere theories and finitary monads in $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$ and discussed the notion of operad, how things go in the enriched context and what changes if you tweak the definitions to allow for more general kinds of limit. We now conclude this iteration of the seminar by bringing to the table “Monads with arities and their associated theories”, by Clemens Berger, Paul-André Melliès and Mark Weber, which answers the (perhaps last) definitional “what-if”: what goes on if you allow for operations of more general arities.

At this point I would like to thank Alexander Campbell, Brendan Fong and Emily Riehl for the amazing organization and support of this seminar, as well as my fellow colleagues, whose posts, presentations and comments drafted a more user-friendly map to traverse this subject.

#### Allowing general arities

Recall that a Lawvere theory can be defined as a pair $<semantics>(I,L)<annotation\; encoding="application/x-tex">(I,L)</annotation></semantics>$, where $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ is a small category with finite coproducts and $<semantics>I:{\aleph}_{0}\to L<annotation\; encoding="application/x-tex">I:\; \backslash aleph\_0\; \backslash to\; L</annotation></semantics>$ is an identity-on-objects finite-coproduct preserving functor. To this data we associate a * nerve functor * $<semantics>{\nu}_{{\aleph}_{0}}:\mathrm{Set}\to \mathrm{PSh}({\aleph}_{0})<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash aleph\_0\}:\; \backslash mathbf\{Set\}\; \backslash to\; PSh(\backslash aleph\_0)</annotation></semantics>$, which takes a set $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$ to its $<semantics>{\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash aleph\_0</annotation></semantics>$-nerve $<semantics>{\nu}_{{\aleph}_{0}}(X):{\aleph}_{0}^{\mathrm{op}}\to \mathrm{Set}<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash aleph\_0\}(X):\; \backslash aleph\_0^\{op\}\; \backslash to\; \backslash mathbf\{Set\}</annotation></semantics>$, the presheaf $<semantics>\mathrm{Set}({i}_{{\aleph}_{0}}(-),X)<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}(i\_\{\backslash aleph\_0\}(-),\; X)</annotation></semantics>$ — the $<semantics>{\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash aleph\_0</annotation></semantics>$-nerve of a set $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$ thus takes a finite cardinal $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$ to $<semantics>{X}^{n}<annotation\; encoding="application/x-tex">X^n</annotation></semantics>$, up to isomorphism. It is easy to check $<semantics>{\nu}_{{\aleph}_{0}}<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash aleph\_0\}</annotation></semantics>$ is faithful, but it is also full, with $<semantics>\alpha \cong {\nu}_{{\aleph}_{0}}({\alpha}_{1})<annotation\; encoding="application/x-tex">\backslash alpha\backslash cong\; \backslash nu\_\{\backslash aleph\_0\}(\backslash alpha\_1)</annotation></semantics>$ for each natural transformation $<semantics>\alpha :{\nu}_{{\aleph}_{0}}(X)\to {\nu}_{{\aleph}_{0}}(X\prime )<annotation\; encoding="application/x-tex">\backslash alpha:\; \backslash nu\_\{\backslash aleph\_0\}(X)\; \backslash to\; \backslash nu\_\{\backslash aleph\_0\}(X\text{\'})</annotation></semantics>$, seeing $<semantics>{\alpha}_{1}<annotation\; encoding="application/x-tex">\backslash alpha\_1</annotation></semantics>$ as a function $<semantics>X\to X\prime <annotation\; encoding="application/x-tex">X\; \backslash to\; X\text{\'}</annotation></semantics>$. This allows us to regard sets as presheaves over the small category $<semantics>{\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash aleph\_0</annotation></semantics>$, and as $<semantics>{\nu}_{{\aleph}_{0}}(X)([n])=\mathrm{Set}([n],X)\cong {X}^{n}<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash aleph\_0\}(X)([n])=\backslash mathbf\{Set\}([n],X)\backslash cong\; X^n</annotation></semantics>$, the $<semantics>{\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash aleph\_0</annotation></semantics>$-nerves can be used to * encode all possible $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$-ary operations on sets*. To capture this behaviour of $<semantics>{\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash aleph\_0</annotation></semantics>$, we are inclined to make the following definition:

**Definition.**
Let $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ be a category and $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ be a full small subcategory of $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$. We say $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ is a * dense generator* of $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ if its associated nerve functor $<semantics>{\nu}_{\mathcal{A}}:\mathcal{C}\to \mathrm{PSh}(\mathcal{A})<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash mathcal\{A\}\}:\; \backslash mathcal\{C\}\; \backslash to\; PSh(\backslash mathcal\{A\})</annotation></semantics>$ is fully faithful, where $<semantics>{\nu}_{\mathcal{A}}(X)=\mathcal{C}({\u0131}_{\mathcal{A}}(-),X)<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash mathcal\{A\}\}(X)=\; \backslash mathcal\{C\}(\backslash imath\_\{\backslash mathcal\{A\}\}(-),\; X)</annotation></semantics>$ for each $<semantics>X\in \mathcal{C}<annotation\; encoding="application/x-tex">X\; \backslash in\; \backslash mathcal\{C\}</annotation></semantics>$.

The idea is that we can replace $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$ and $<semantics>{\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash aleph\_0</annotation></semantics>$ in the original definition of Lawvere theory by a category $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ with a dense generator $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$. This allows us to have operations with arities more diverse than simply finite cardinals, while still retaining “good behaviour” — if we think about the dense generator as giving the “allowed arities”, we end up being able to extend all the previous concepts and make the following analogies:

$$<semantics>\begin{array}{l}\\ \text{Lawvere theory}\phantom{\rule{thinmathspace}{0ex}}L& \text{Theory}\phantom{\rule{thinmathspace}{0ex}}\Theta \phantom{\rule{thinmathspace}{0ex}}\text{with arity}\phantom{\rule{thinmathspace}{0ex}}\mathcal{A}\\ \text{Model of}\phantom{\rule{thinmathspace}{0ex}}{L}^{\mathrm{op}}& \Theta \text{-model}\\ \text{Finitary monad}& \text{Monad with arity}\phantom{\rule{thinmathspace}{0ex}}\mathcal{A}\\ \text{Globular (Batanin) operad}& \text{Homogenous globular theory}\\ \text{Symmetric operad}& \Gamma \text{-homogeneous theory}\\ & \end{array}annotation\; encoding="application/x-tex"\; \backslash array\; \{\backslash arrayopts\{\; \backslash colalign\{left\; left\}\; \backslash rowlines\{solid\}\; \}\; \backslash \backslash \; \backslash text\{Lawvere\; theory\}\backslash ,\; L\; \backslash text\{Theory\}\backslash ,\; \backslash Theta\; \backslash ,\; \backslash text\{with\; arity\}\; \backslash ,\; \backslash mathcal\{A\}\backslash \backslash \; \backslash text\{Model\; of\}\backslash ,\; L^\{op\}\; \backslash Theta\backslash text\{-model\}\backslash \backslash \; \backslash text\{Finitary\; monad\}\backslash text\{Monad\; with\; arity\}\backslash ,\; \backslash mathcal\{A\}\backslash \backslash \; \backslash text\{Globular\; (Batanin)\; operad\}\backslash text\{Homogenous\; globular\; theory\}\backslash \backslash \; \backslash text\{Symmetric\; operad\}\backslash Gamma\backslash text\{-homogeneous\; theory\}\backslash \backslash \; nbsp;\; nbsp;\; \}\; /annotation/semantics$$ We’ll now discuss each generalised concept and important/useful properties.

If $<semantics>(I,L)<annotation\; encoding="application/x-tex">(I,L)</annotation></semantics>$ is a Lawvere theory, the restriction functor $<semantics>{I}^{*}:\mathrm{PSh}(L)\to \mathrm{PSh}({\aleph}_{0})<annotation\; encoding="application/x-tex">I^\{\backslash ast\}:\; PSh(L)\; \backslash to\; PSh(\backslash aleph\_0)</annotation></semantics>$ induces a monad $<semantics>{I}^{*}{I}_{!}<annotation\; encoding="application/x-tex">I^\{\backslash ast\}\; I\_!</annotation></semantics>$, where $<semantics>{I}_{!}<annotation\; encoding="application/x-tex">I\_!</annotation></semantics>$ is left Kan extension along $<semantics>I<annotation\; encoding="application/x-tex">I</annotation></semantics>$. This monad preserves the essential image of the nerve functor $<semantics>{\nu}_{{\aleph}_{0}}<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash aleph\_0\}</annotation></semantics>$, and in fact this condition reduces to preservation of coproducts by $<semantics>I<annotation\; encoding="application/x-tex">I</annotation></semantics>$ (refer to 3.5 in the paper for further details). If $<semantics>M<annotation\; encoding="application/x-tex">M</annotation></semantics>$ is a model of $<semantics>{L}^{\mathrm{op}}<annotation\; encoding="application/x-tex">L^\{op\}</annotation></semantics>$ on $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$ in the usual sense (i.e $<semantics>M:{L}^{\mathrm{op}}\to \mathrm{Set}<annotation\; encoding="application/x-tex">M:\; L^\{op\}\; \backslash to\; \backslash mathbf\{Set\}</annotation></semantics>$ preserves finite products) we can see that its restriction along $<semantics>I<annotation\; encoding="application/x-tex">I</annotation></semantics>$ is isomorphic to the $<semantics>{\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash aleph\_0</annotation></semantics>$-nerve of $<semantics>\mathrm{MI}([1])<annotation\; encoding="application/x-tex">MI([1])</annotation></semantics>$ by arguing that

$$<semantics>({I}^{*}M)[n]=\mathrm{MI}[n]=M{\underset{\u23df}{(\coprod _{n}I[1])}}_{\text{in}\phantom{\rule{thinmathspace}{0ex}}L}=M{\underset{\u23df}{(\prod _{n}I[1])}}_{\text{in}\phantom{\rule{thinmathspace}{0ex}}{L}^{\mathrm{op}}}\cong \prod _{n}\mathrm{MI}[1]\cong \mathrm{MI}[1{]}^{n}\cong {\nu}_{{\aleph}_{0}}(\mathrm{MI}[1])[n],<annotation\; encoding="application/x-tex">\; (I^\{\backslash ast\}\; M)[n]\; =\; MI[n]\; =\; M\; \backslash underbrace\{(\backslash coprod\_n\; I[1])\}\_\{\backslash text\{in\}\; \backslash ,\; L\}\; =\; M\backslash underbrace\{(\backslash prod\_n\; I[1])\}\_\{\backslash text\{in\}\backslash ,\; L^\{op\}\}\backslash cong\; \backslash prod\_n\; MI[1]\; \backslash cong\; MI[1]^n\; \backslash cong\; \backslash nu\_\{\backslash aleph\_0\}(MI[1])[n],\; </annotation></semantics>$$

and so we may want to define:

**Definition.**
Let $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ be a category with a dense generator $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$. A * theory with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ * on $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ is a pair $<semantics>(\Theta ,j)<annotation\; encoding="application/x-tex">(\backslash Theta,j)</annotation></semantics>$, where $<semantics>j:\mathcal{A}\to \Theta <annotation\; encoding="application/x-tex">j:\; \backslash mathcal\{A\}\; \backslash to\; \backslash Theta</annotation></semantics>$ is a bijective-on-objects functor such that the induced monad $<semantics>{j}^{*}{j}_{!}<annotation\; encoding="application/x-tex">j^\{\backslash ast\}j\_!</annotation></semantics>$ on $<semantics>\mathrm{PSh}(\mathcal{A})<annotation\; encoding="application/x-tex">PSh(\backslash mathcal\{A\})</annotation></semantics>$ preserves the essential image of the associated nerve functor $<semantics>{\nu}_{\mathcal{A}}<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash mathcal\{A\}\}</annotation></semantics>$. A * $<semantics>\Theta <annotation\; encoding="application/x-tex">\backslash Theta</annotation></semantics>$-model* is a presheaf on $<semantics>\Theta <annotation\; encoding="application/x-tex">\backslash Theta</annotation></semantics>$ whose restriction along $<semantics>j<annotation\; encoding="application/x-tex">j</annotation></semantics>$ is isomorphic to some $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$-nerve.

Again, for $<semantics>\mathcal{A}={\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}=\backslash aleph\_0</annotation></semantics>$, this requirement on models says a $<semantics>\Theta <annotation\; encoding="application/x-tex">\backslash Theta</annotation></semantics>$-model $<semantics>M<annotation\; encoding="application/x-tex">M</annotation></semantics>$ restricts to powers of some object: $<semantics>{I}^{*}M(-)=\mathrm{MI}(-)\cong {X}^{|-|}<annotation\; encoding="application/x-tex">I^\backslash ast\; M(-)=MI(-)\; \backslash cong\; X^\{|-|\}</annotation></semantics>$ for some set $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$, the outcome we wanted for models of Lawvere theories.

A * morphism of models* is still just a natural transformation between them as presheaves and a * morphism of theories* $<semantics>({\Theta}_{1},{j}_{1})\to ({\Theta}_{2},{j}_{2})<annotation\; encoding="application/x-tex">(\backslash Theta\_1,\; j\_1)\; \backslash to\; (\backslash Theta\_2,\; j\_2)</annotation></semantics>$ is a functor $<semantics>\theta :{\Theta}_{1}\to {\Theta}_{2}<annotation\; encoding="application/x-tex">\backslash theta:\; \backslash Theta\_1\; \backslash to\; \backslash Theta\_2</annotation></semantics>$ that intertwines with the arity functors, i.e $<semantics>{j}_{2}=\theta {j}_{1}<annotation\; encoding="application/x-tex">j\_2=\backslash theta\; j\_1</annotation></semantics>$. We’ll write $<semantics>\mathrm{Mod}(\Theta )<annotation\; encoding="application/x-tex">Mod(\backslash Theta)</annotation></semantics>$ for the full subcategory of $<semantics>\mathrm{PSh}(\Theta )<annotation\; encoding="application/x-tex">PSh(\backslash Theta)</annotation></semantics>$ consisting of the models of $<semantics>\Theta <annotation\; encoding="application/x-tex">\backslash Theta</annotation></semantics>$ and $<semantics>\mathrm{Th}(\mathcal{C},\mathcal{A})<annotation\; encoding="application/x-tex">Th(\backslash mathcal\{C\},\; \backslash mathcal\{A\})</annotation></semantics>$ for the category of theories with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ on $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$. We aim to prove a result that establishes an equivalence between $<semantics>\mathrm{Th}(\mathcal{C},\mathcal{A})<annotation\; encoding="application/x-tex">Th(\backslash mathcal\{C\},\backslash mathcal\{A\})</annotation></semantics>$ and some category of monads, to mirror the situation between Lawvere theories and finitary monads on $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$.

#### Dense generators and nerves

Having a dense generator is desirable because we can then mimic the following situation:

Recall that if $<semantics>\mathcal{D}<annotation\; encoding="application/x-tex">\backslash mathcal\{D\}</annotation></semantics>$ is small and $<semantics>F:\mathcal{D}\to \mathrm{Set}<annotation\; encoding="application/x-tex">F:\backslash mathcal\{D\}\; \backslash to\; \backslash mathbf\{Set\}</annotation></semantics>$ is a functor, then we can form a diagram of shape $<semantics>(*\downarrow F{)}^{\mathrm{op}}<annotation\; encoding="application/x-tex">(\{\backslash ast\}\backslash downarrow\; F)^\{op\}</annotation></semantics>$ over $<semantics>[\mathcal{D},\mathrm{Set}]<annotation\; encoding="application/x-tex">[\backslash mathcal\{D\},\; \backslash mathbf\{Set\}]</annotation></semantics>$ by composing the (opposite) of the natural projection functor $<semantics>(*\downarrow F)\to \mathcal{D}<annotation\; encoding="application/x-tex">(\{\backslash ast\}\backslash downarrow\; F)\; \backslash to\; \backslash mathcal\{D\}</annotation></semantics>$ and the Yoneda embedding. We may then consider the cocone

$$<semantics>\mu =({\mu}_{(d,x)}={\mu}_{x}:\mathcal{D}(d,-)\to F\mid (d,x)\in (*\downarrow F{)}^{\mathrm{op}}),<annotation\; encoding="application/x-tex">\; \backslash mu=(\backslash mu\_\{(d,x)\}=\backslash mu\_x:\; \backslash mathcal\{D\}(d,-)\; \backslash to\; F\; \backslash mid\; (d,x)\; \backslash in\; (\{\backslash ast\}\; \backslash downarrow\; F)^\{op\}),</annotation></semantics>$$

where $<semantics>{\mu}_{x}<annotation\; encoding="application/x-tex">\backslash mu\_x</annotation></semantics>$ is the natural transformation corresponding to $<semantics>x\in F(d)<annotation\; encoding="application/x-tex">x\; \backslash in\; F(d)</annotation></semantics>$ via the Yoneda lemma, and find out it is actually a colimit, * canonically expressing $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ as a colimit of representable functors* — if you are so inclined, you might want to look at this as the coend identity

$$<semantics>F(-)={\int}^{d\in \mathcal{D}}F(d)\times \mathcal{D}(-,d)<annotation\; encoding="application/x-tex">\; F(-)=\; \backslash int^\{d\; \backslash in\; \backslash mathcal\{D\}\}\; F(d)\; \backslash times\; \backslash mathcal\{D\}(-,d)\; </annotation></semantics>$$

when $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ is a presheaf on $<semantics>\mathcal{D}<annotation\; encoding="application/x-tex">\backslash mathcal\{D\}</annotation></semantics>$. Likewise for $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$ an object of a category $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ with dense generator $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$, there is an associated diagram $<semantics>{a}_{X}:\mathcal{A}/X\to \mathcal{C}<annotation\; encoding="application/x-tex">a\_X:\; \backslash mathcal\{A\}/X\; \backslash to\; \backslash mathcal\{C\}</annotation></semantics>$, which comes equipped with an obvious natural transformation to the constant functor on $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$, whose $<semantics>(A\stackrel{f}{\to}X)<annotation\; encoding="application/x-tex">(A\; \backslash xrightarrow\{f\}\; X)</annotation></semantics>$-component is simply $<semantics>f<annotation\; encoding="application/x-tex">f</annotation></semantics>$ itself — this is called the $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$-cocone over $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$, and it is just the cocone of vertex $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$ under the diagram $<semantics>{a}_{X}<annotation\; encoding="application/x-tex">a\_X</annotation></semantics>$ of shape $<semantics>\mathcal{A}/X<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}/X</annotation></semantics>$ in $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ whose legs consist of all morphisms $<semantics>A\to X<annotation\; encoding="application/x-tex">A\; \backslash to\; X</annotation></semantics>$ with $<semantics>A\in \mathcal{A}<annotation\; encoding="application/x-tex">A\; \backslash in\; \backslash mathcal\{A\}</annotation></semantics>$. Note that if $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ is small (as is the case), then this diagram is small and, if $<semantics>\mathcal{C}=\mathrm{PSh}(\mathcal{A})<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}=PSh(\backslash mathcal\{A\})</annotation></semantics>$, the slice category $<semantics>\mathcal{A}/X<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}/X</annotation></semantics>$ reduces to the category of elements of the presheaf $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$ and this construction gives the Yoneda cocone under $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$. One can show that

**Proposition.** A small full subcategory $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ of $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ is a dense generator precisely when the $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$-cocones are actually colimit-cocones in $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$.

This canonically makes * every object $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$ of $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ a colimit of objects in $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ *, and in view of this result it makes sense to define:

**Definition.**
Let $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ be a category with a dense generator $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$. A monad $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ on $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ is a *monad with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$* when $<semantics>{\nu}_{\mathcal{A}}T<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash mathcal\{A\}\}T</annotation></semantics>$ takes the $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$-cocones of $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ to colimit-cocones in $<semantics>\mathrm{PSh}(\mathcal{A})<annotation\; encoding="application/x-tex">PSh(\backslash mathcal\{A\})</annotation></semantics>$.

That is, the monad has arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ whenever scrambling the nerve functor by first applying $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ does not undermine its capacity of turning $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$-cocones into colimits, which in turns preserves the status of $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ as a dense generator, morally speaking — the *Nerve Theorem* makes this statement more precise:

**The Nerve Theorem.** Let $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ be a category with a dense generator $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$. For any monad $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$, the full subcategory $<semantics>{\Theta}_{T}<annotation\; encoding="application/x-tex">\backslash Theta\_T</annotation></semantics>$ spanned by the free $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$-algebras on objects of $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ is a dense generator of the Eilenberg-Moore category $<semantics>{\mathcal{C}}^{T}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}^T</annotation></semantics>$. The essential image of the associated nerve functor is spanned by those presheaves whose restriction along $<semantics>{j}_{T}<annotation\; encoding="application/x-tex">j\_T</annotation></semantics>$ belongs to the essential image of the $<semantics>{\nu}_{\mathcal{A}}<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash mathcal\{A\}\}</annotation></semantics>$, where $<semantics>{j}_{T}:\mathcal{A}\to {\Theta}_{T}<annotation\; encoding="application/x-tex">j\_T:\; \backslash mathcal\{A\}\; \backslash to\; \backslash Theta\_T</annotation></semantics>$ is obtained by restricting the free $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$-algebra functor.

The proof given relies on an equivalent characterization for monads with arities, namely that a monad $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ on a category $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ is a monad with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ if and only if the “generalised lifting (pseudocommutative) diagram”

$$<semantics>\begin{array}{ccc}{\mathcal{C}}^{T}& \stackrel{{\nu}_{T}}{\u27f6}& \mathrm{PSh}({\Theta}_{T})\\ {}_{U}\downarrow & & {\downarrow}_{{j}_{T}^{*}}\\ \mathcal{C}& \underset{{\nu}_{\mathcal{A}}}{\u27f6}& \mathrm{PSh}(\mathcal{A})\\ \end{array}.<annotation\; encoding="application/x-tex">\; \backslash begin\{matrix\}\; \backslash mathcal\{C\}^T\; \&\; \backslash overset\{\backslash nu\_T\}\{\backslash longrightarrow\}\; \&\; PSh(\backslash Theta\_T)\; \backslash \backslash \; \{\}\_\{U\}\backslash downarrow\; \&\&\; \backslash downarrow\_\{j\_T^\{\backslash ast\}\}\; \backslash \backslash \; \backslash mathcal\{C\}\; \&\; \backslash underset\{\backslash nu\_\{\backslash mathcal\{A\}\}\}\{\backslash longrightarrow\}\; \&\; PSh(\backslash mathcal\{A\})\; \backslash \backslash \; \backslash end\{matrix\}.\; </annotation></semantics>$$

is an exact adjoint square, meaning the mate $<semantics>({j}_{T}{)}_{!}{\nu}_{\mathcal{A}}\Rightarrow {\nu}_{T}F<annotation\; encoding="application/x-tex">(j\_T)\_!\backslash nu\_\{\backslash mathcal\{A\}\}\; \backslash Rightarrow\; \backslash nu\_T\; F</annotation></semantics>$ of the invertible $<semantics>2<annotation\; encoding="application/x-tex">2</annotation></semantics>$-cell implicit in the above square is also invertible, where $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ is the free $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$-algebra functor. Note $<semantics>{j}_{T}^{*}<annotation\; encoding="application/x-tex">j\_T^\{\backslash ast\}</annotation></semantics>$ is monadic, so this diagram indeed gives some sort of lifting of the nerve functor on $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ to the level of monad algebras.

We can build on this result a little bit. Let $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$ be a regular cardinal (at this point you might want to check David’s discussion on finite presentability).

**Definition.** A category $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ is *$<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-accessible* if it has $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered colimits and a dense generator $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ comprised only of $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-presentable objects such that $<semantics>\mathcal{A}/X<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}/X</annotation></semantics>$ is $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered for each object $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$ of $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$. If in addition the category is cocomplete, we say it is *locally $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-presentable*.

If $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ is $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-accessible, there is a god-given choice of dense generator — we take $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ to be a skeleton of the full subcategory $<semantics>\mathcal{C}(\alpha )<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}(\backslash alpha)</annotation></semantics>$ spanned by the $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-presentable objects of $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$. As all objects in $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ are $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-presentable, the associated nerve functor preserves $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered colimits and so any monad $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ preserving $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered colimits is a monad with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$. The essential image of $<semantics>{\nu}_{\mathcal{A}}<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash mathcal\{A\}\}</annotation></semantics>$ is spanned by the $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-flat presheaves on $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ (meaning presheaves whose categories of elements are $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered). As a consequence, any given object in an $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-accessible category is canonically an $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered colimit of $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-presentable objects and we can prove:

**Theorem (Gabriel-Ulmer, Adámek-Rosický).**
If a monad $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ on an $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-accessible category $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ preserves $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered colimits, then its category of algebras $<semantics>{\mathcal{C}}^{T}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}^T</annotation></semantics>$ is $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-accessible, with a dense generator $<semantics>{\Theta}_{T}<annotation\; encoding="application/x-tex">\backslash Theta\_T</annotation></semantics>$ spanned by the free $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$-algebras on (a skeleton $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$ of) the $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-presentable objects $<semantics>C(\alpha )<annotation\; encoding="application/x-tex">C(\backslash alpha)</annotation></semantics>$. Moreover, this category of algebras is equivalent to the full subcategory of $<semantics>\mathrm{PSh}({\Theta}_{T})<annotation\; encoding="application/x-tex">PSh(\backslash Theta\_T)</annotation></semantics>$ spanned by those presheaves whose restriction along $<semantics>{j}_{T}<annotation\; encoding="application/x-tex">j\_T</annotation></semantics>$ is $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-flat.

**Proof.** We know $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ is a monad with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$. That $<semantics>{\Theta}_{T}<annotation\; encoding="application/x-tex">\backslash Theta\_T</annotation></semantics>$ is a dense generator as stated follows from its definition and the Nerve Theorem. Now, $<semantics>{\mathcal{C}}^{T}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}^T</annotation></semantics>$ has $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered colimits since $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ has and $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ preserves them. As the forgetful functor $<semantics>U:{\mathcal{C}}^{T}\to \mathcal{C}<annotation\; encoding="application/x-tex">U:\; \backslash mathcal\{C\}^T\; \backslash to\; \backslash mathcal\{C\}</annotation></semantics>$ preserves $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered colimits (a monadic functor creates all colimits $<semantics>\mathcal{C}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}</annotation></semantics>$ has and $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ preserves), it follows that the free algebra functor preserves $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-presentability: $<semantics>\mathcal{C}(\mathrm{FA},-)\cong \mathcal{C}(A,U(-))<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}(FA,-)\; \backslash cong\; \backslash mathcal\{C\}(A,\; U(-))</annotation></semantics>$ preserves $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered colimits whenever $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-presentable, and so objects of $<semantics>{\Theta}_{T}<annotation\; encoding="application/x-tex">\backslash Theta\_T</annotation></semantics>$ are $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-presentable. One can then check each $<semantics>\mathcal{A}/X<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}/X</annotation></semantics>$ is $<semantics>\alpha <annotation\; encoding="application/x-tex">\backslash alpha</annotation></semantics>$-filtered. $<semantics>\square <annotation\; encoding="application/x-tex">\backslash square</annotation></semantics>$

Note that this theorem says, for $<semantics>\alpha ={\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash alpha=\backslash aleph\_0</annotation></semantics>$, that if a monad on sets is finitary, then its category of algebras (i.e models for the associated classical Lawvere theory) is accessible, with a dense generator given by all the free $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$-algebras on finite sets: this is because a finitely-presentable (i.e $<semantics>{\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash aleph\_0</annotation></semantics>$-presentable) set is precisely the same as a finite set. As a consequence, the typical “algebraic gadgets” are canonically a colimit of free ones on finitely many generators.

#### Theories and monads (with arities) are equivalent

If $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$ is a monad with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$, then $<semantics>({\Theta}_{T},{j}_{T})<annotation\; encoding="application/x-tex">(\backslash Theta\_T,\; j\_T)</annotation></semantics>$ is a theory with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$. The Nerve Theorem then guarantees that $<semantics>{\nu}_{T}:{\mathcal{C}}^{T}\to \mathrm{PSh}({\Theta}_{T})<annotation\; encoding="application/x-tex">\backslash nu\_T:\; \backslash mathcal\{C\}^T\; \backslash to\; PSh(\backslash Theta\_T)</annotation></semantics>$ induces an equivalence of categories between $<semantics>{\Theta}_{T}<annotation\; encoding="application/x-tex">\backslash Theta\_T</annotation></semantics>$-models and $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$-algebras, since its essential image is, by definition, the category of $<semantics>{\Theta}_{T}<annotation\; encoding="application/x-tex">\backslash Theta\_T</annotation></semantics>$-models and the functor is fully faithful. This gives us hope that the situation with Lawvere theories and finitary monads can be extended, and this is indeed the case: the assignment $<semantics>T\mapsto ({\Theta}_{T},{j}_{T})<annotation\; encoding="application/x-tex">T\; \backslash mapsto\; (\backslash Theta\_T,\; j\_T)</annotation></semantics>$ extends to a functor $<semantics>\mathrm{Mnd}(\mathcal{C},\mathcal{A})\to \mathrm{Th}(\mathcal{C},\mathcal{A})<annotation\; encoding="application/x-tex">\backslash mathbf\{Mnd\}(\backslash mathcal\{C\},\; \backslash mathcal\{A\})\; \backslash to\; \backslash mathbf\{Th\}(\backslash mathcal\{C\},\; \backslash mathcal\{A\})</annotation></semantics>$, which forms an equivalence of categories together with the functor $<semantics>\mathrm{Th}(\mathcal{C},\mathcal{A})\to \mathrm{Mnd}(\mathcal{C},\mathcal{A})<annotation\; encoding="application/x-tex">\backslash mathbf\{Th\}(\backslash mathcal\{C\},\; \backslash mathcal\{A\})\; \backslash to\; \backslash mathbf\{Mnd\}(\backslash mathcal\{C\},\; \backslash mathcal\{A\})</annotation></semantics>$ that takes a theory $<semantics>(\Theta ,j)<annotation\; encoding="application/x-tex">(\backslash Theta,\; j)</annotation></semantics>$ to the monad $<semantics>{\rho}_{\mathcal{A}}T{\nu}_{\mathcal{A}}<annotation\; encoding="application/x-tex">\backslash rho\_\{\backslash mathcal\{A\}\}T\backslash nu\_\{\backslash mathcal\{A\}\}</annotation></semantics>$ with arities $<semantics>\mathcal{A}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}</annotation></semantics>$, where $<semantics>{\rho}_{\mathcal{A}}<annotation\; encoding="application/x-tex">\backslash rho\_\{\backslash mathcal\{A\}\}</annotation></semantics>$ is a choice of right adjoint to $<semantics>{\nu}_{\mathcal{A}}:\mathcal{C}\to \mathrm{EssIm}({\nu}_{\mathcal{A}})<annotation\; encoding="application/x-tex">\backslash nu\_\{\backslash mathcal\{A\}\}:\; \backslash mathcal\{C\}\; \backslash to\; EssIm(\backslash nu\_\{\backslash mathcal\{A\}\})</annotation></semantics>$. When $<semantics>\mathcal{C}=\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathcal\{C\}=\backslash mathbf\{Set\}</annotation></semantics>$ and $<semantics>\mathcal{A}={\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash mathcal\{A\}=\backslash aleph\_0</annotation></semantics>$, we recover the Lawvere theory/finitary monad equivalence.

#### Relation to operads and examples

Certain kinds of theories with arities are equivalent to operads. Namely, there is a notion of *homogeneous globular theory* that corresponds to globular (Batanin) operads. Similarly, there is a notion of $<semantics>\Gamma <annotation\; encoding="application/x-tex">\backslash Gamma</annotation></semantics>$-homogeneous theory that corresponds to symmetric operads. The remainder of the paper brings other equivalent definitions for monad with arities and builds a couple of examples, such as the free groupoid monad, which is a monad with arities given by (finite, connected) acyclic graphs. A notable example is that dagger categories arise as models of a theory on involutive graphs with non-trivial arities.

## May 11, 2017

### ZapperZ - Physics and Physicists

Almost half of the degree holders left school to go into the workforce, with about 54% going on to graduate school. This is a significant percentage, and as educators, we need to make sure we prepare physics graduates for such a career path and not assume that they will all go on to graduate schools. This means that we design a program in which they have valuable and usable skills by the time they graduate.

Zz.

## May 10, 2017

### ZapperZ - Physics and Physicists

His dad explained that his son 'likes to be the life of the party, which gets him in trouble from time to time.''For some reason I said, "hey, if we get another call I'm going to show up in school and sit beside you in class,"' he said.Unfortunately for the 17-year-old, that call did come.

### Tommaso Dorigo - Scientificblogging

Yesterday I visited the Liceo “Benedetti” of Venice, where 40 students are preparing their artwork for a project of communicating science with art that will culminate in an exhibit at the Palazzo del Casinò of the Lido of Venice, during the week of the EPS conference in July.

## May 09, 2017

### Symmetrybreaking - Fermilab/SLAC

Linac 4 will replace an older accelerator as the first step in the complex that includes the LHC.

At a ceremony today, CERN European research center inaugurated its newest accelerator.

Linac 4 will eventually become the first step in CERN’s accelerator chain, delivering proton beams to a wide range of experiments, including those at the Large Hadron Collider.

After an extensive testing period, Linac 4 will be connected to CERN’s accelerator complex during a long technical shutdown in 2019-20. Linac 4 will replace Linac 2, which was put into service in 1978. Linac 4 will feed the CERN accelerator complex with particle beams of higher energy.

“We are delighted to celebrate this remarkable accomplishment,” says CERN Director General Fabiola Gianotti. “Linac 4 is a modern injector and the first key element of our ambitious upgrade program, leading to the High-Luminosity LHC. This high-luminosity phase will considerably increase the potential of the LHC experiments for discovering new physics and measuring the properties of the Higgs particle in more detail.”

“This is an achievement not only for CERN, but also for the partners from many countries who contributed in designing and building this new machine,” says CERN Director for Accelerators and Technology Frédérick Bordry. “We also today celebrate and thank the wide international collaboration that led this project, demonstrating once again what can be accomplished by bringing together the efforts of many nations.”

The linear accelerator is the first essential element of an accelerator chain. In the linear accelerator, the particles are produced and receive the initial acceleration. The density and intensity of the particle beams are also shaped in the linac. Linac 4 is an almost 90-meter-long machine sitting 12 meters below the ground. It took nearly 10 years to build it.

Linac 4 will send negative hydrogen ions, consisting of a hydrogen atom with two electrons, to CERN’s Proton Synchrotron Booster, which further accelerates the negative ions and removes the electrons. Linac 4 will bring the beam up to an energy of 160 million electronvolts, more than 3 times the energy of its predecessor. The increase in energy, together with the use of hydrogen ions, will enable doubling the beam intensity delivered to the LHC, contributing to an increase in the luminosity of the LHC by 2021.

Luminosity is a parameter indicating the number of particles colliding within a defined amount of time. The peak luminosity of the LHC is planned to be increased by a factor of 5 by the year 2025. This will make it possible for the experiments to accumulate about 10 times more data over the period 2025 to 2035 than before.

*Editor's note: This article is based on a CERN press release.*

### Symmetrybreaking - Fermilab/SLAC

The authors of *We Have No Idea *remind us that there are still many unsolved mysteries in science.

What is dark energy? Why aren’t we made of antimatter? How many dimensions are there?

These are a few of the many unanswered questions that Jorge Cham, creator of the online comic Piled Higher and Deeper, and Daniel Whiteson, an experimental particle physicist at the University of California, Irvine, explain in their new book, We Have No Idea. In the process, they remind readers of one key point: When it comes to our universe, there’s a lot we still don’t know.

The duo started working together in 2008 after Whiteson reached out to Cham, asking if he’d be willing to help create physics cartoons. “I always thought physics was well connected to the way comics work,” Whiteson says. “Because, what’s a Feynman diagram but a little cartoon of particles hitting each other?” (Feynman diagrams are pictures commonly used in particle physics papers that represent the interactions of subatomic particles.)

Before working on this book, the pair made a handful of popular YouTube videos on topics like dark matter, extra dimensions and the Higgs boson. Many of these subjects are also covered in We Have No Idea.

One of the main motivators of this latest project was to address a “certain apathy toward science,” Cham says. “I think we both came into it having this feeling that the general public either thinks scientists have everything figured out, or they don't really understand what scientists are doing.”

To get at this issue, the pair focused on topics that even someone without a science background could find compelling. “You don’t need 10 years of physics background to know [that] questions about how the universe started or what it’s made of are interesting,” Whiteson says. “We tried to find questions that were gut-level approachable.”

Another key theme of the book, the authors say, is the line between what science can and cannot tell us. While some of the possible solutions to the universe’s mysteries have testable predictions, others (such as string theory) currently do not. “We wanted questions that were accessible yet answerable,” says Whiteson. “We wanted to show people that there were deep, basic, simple questions that we all had, but that the answers were out there.”

Many scientists are hard at work trying to fill the gaping holes in our knowledge about the universe. Particle physicists, for example, are exploring a number of these questions, such as those about the nature of antimatter and mass.

Some lines of inquiry have brought different research communities together. Dark matter searches, for example, were primarily the realm of cosmologists, who probe large-scale structures of the universe. However, as the focus shifted to finding out what particle—or particles—dark matter was made of, this area of study started to attract astrophysicists as well.

Why are people trying to answer these questions? “I think science is an expression of humanity and our curiosity to know the answers to basic questions we ask ourselves: Who are we? Why are we here? How does the world work?” Whiteson says. “On the other hand, questions like these lead to understanding, and understanding leads to being able to have greater power over the environment to solve our problems.

In the very last chapter of the book, the authors explain the idea of a “testable universe,” or the parts of the universe that fall within the bounds of science. In the Stone Ages, when humans had very few tools at their disposal, the testable universe was very small. But it increased as people built telescopes, satellites and particle colliders, and it continues to expand with ongoing advances in science and technology. “That’s the exciting thing,” Cham says. “Our ability to answer these questions is growing.”

Some mysteries of the universe still live in the realm of philosophy. But tomorrow, next year or a thousand years from now, a scientist may come along and devise an experiment that will be able to find the answers.

“We’re in a special place in history when most of the world seems explained,” Whiteson says. Thousands of years ago, basic questions, such as why fire burns or where rain comes from, were still largely a mystery. “These days, all those mysteries seem answered, but the truth is, there’s a lot of mysteries left. [If] you want to make a massive imprint on human intellectual history, there’s plenty of room for that.”

## May 06, 2017

### The n-Category Cafe

*Guest post by Daniel Cicala*

The Kan Extension Seminar II continues with a discussion of the paper Notions of Lawvere Theory by Stephen Lack and Jirí Rosický.

In his landmark thesis, William Lawvere introduced a method to the study of universal algebra that was vastly more abstract than those previously used. This method actually turns certain mathematical stuff, structure, and properties into a mathematical object! This is achieved with a *Lawvere theory*: a bijective-on-objects product preserving functor $<semantics>T:{\aleph}_{0}^{\text{op}}\to L<annotation\; encoding="application/x-tex">T\; \backslash colon\; \backslash aleph^\{\backslash text\{op\}\}\_0\; \backslash to\; \backslash mathbf\{L\}</annotation></semantics>$ where $<semantics>{\aleph}_{0}<annotation\; encoding="application/x-tex">\backslash aleph\_0</annotation></semantics>$ is a skeleton of the category $<semantics>\mathrm{FinSet}<annotation\; encoding="application/x-tex">\backslash mathbf\{FinSet\}</annotation></semantics>$ and $<semantics>L<annotation\; encoding="application/x-tex">\backslash mathbf\{L\}</annotation></semantics>$ is a category with finite products. The analogy between algebraic gadgets and Lawvere theories reads as: stuff, structure, and properties correspond respectively to 1, morphisms, and commuting diagrams.

To get an actual instance, or a *model*, of an algebraic gadget from a Lawvere theory, we take a product preserving functor $<semantics>m:T\to \mathrm{Set}<annotation\; encoding="application/x-tex">m\; \backslash colon\; \backslash mathbf\{T\}\; \backslash to\; \backslash mathbf\{Set\}</annotation></semantics>$. A model picks out a set $<semantics>m(1)<annotation\; encoding="application/x-tex">m(1)</annotation></semantics>$ and $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$-ary operations $<semantics>m(f):m(1{)}^{n}\to m(1)<annotation\; encoding="application/x-tex">m(f)\; \backslash colon\; m(1)^n\; \backslash to\; m(1)</annotation></semantics>$ for every $<semantics>T<annotation\; encoding="application/x-tex">T</annotation></semantics>$-morphism $<semantics>f:n\to 1<annotation\; encoding="application/x-tex">f\; \backslash colon\; n\; \backslash to\; 1</annotation></semantics>$.

To read more about classical Lawvere theories, you can read Evangelia Aleiferi’s discussion of Hyland and Power’s paper on the topic.

With this elegant perspective on universal algebra, we do what mathematicians are wont to do: generalize it. However, there is much to consider undertaking such a project. Firstly, what elements of the theory ought to be generalized? Lack and Rosický provide a clear answer to this question. They generalize along the following three tracks:

consider a class of limits besides finite products,

replace the base category $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$ with some other suitable category, and

enrich everything.

Another important consideration is to determine exactly how far to generalize. Why not just go as far as possible? Here are two reasons. First, there are a number of results in this paper that stand up to further generalization if one doesn’t care about constructibility. A second limiting factor of generalization is that one should ensure that central properties still hold. In *Notions of Lawvere Theory*, the properties lifted from classical Lawvere theories are

the correspondence between Lawvere theories and monads,

that algebraic functors have left adjoints, and

models form reflective subcategories of certain functor categories.

Before starting the discussion of the paper, I would like to take a moment to thank Alexander, Brendan and Emily for running this seminar. I have truly learned a lot and have enjoyed wonderful conversations with everyone involved.

## Replacing finite limits

To find a suitable class of limits to replace finite products, we require the concept of presentability. The best entry point is to learn about local finite presentability, which David Myers has discussed here. With a little modification to the ideas there, we define a notion of local strong finite presentability and local $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-presentability for a class of limits $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$.

We begin with *sifted colimits*, which are those $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$-valued colimits that commute with finite products. Note the familiarity of this definition with the commutativity property of filtered colimits. Of course, filtered colimits are also sifted. Another example is a reflexive pair, that is a category with shape

Anyway, we now look at the *strongly finitely presentable objects* in a category $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$. These are those objects $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ whose representable $<semantics>C(x,-):C\to \mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}(x,-)\; \backslash colon\; \backslash mathbf\{C\}\; \backslash to\; \backslash mathbf\{Set\}</annotation></semantics>$ preserves sifted colimits. Denote the full subcategory of these by $<semantics>{C}_{\text{sfp}}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}\_\{\backslash text\{sfp\}\}</annotation></semantics>$. Some simple examples include $<semantics>{\mathrm{Set}}_{\text{sfp}}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}\_\{\backslash text\{sfp\}\}</annotation></semantics>$, which consists of the finite sets, and $<semantics>{\mathrm{Ab}}_{\text{sfp}}<annotation\; encoding="application/x-tex">\backslash mathbf\{Ab\}\_\{\backslash text\{sfp\}\}</annotation></semantics>$, which has the free and finitely generated Abelian groups. Also, given a category $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$ of models for a Lawvere theory, $<semantics>{C}_{\text{sfp}}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}\_\{\backslash text\{sfp\}\}</annotation></semantics>$ is exactly those finitely presentable objects that are regular projective. A category $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$ is *locally strongly finitely presentable* if it is cocomplete, $<semantics>{C}_{\text{sfp}}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}\_\{\backslash text\{sfp\}\}</annotation></semantics>$ is small, and any $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$-object is a sifted colimit of a diagram in $<semantics>{C}_{\text{sfp}}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}\_\{\backslash text\{sfp\}\}</annotation></semantics>$. There is also a nice characterization (Theorem 3.1 in the paper) that states $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$ is locally strongly finitely presentable if and only if $<semantics>{C}_{\text{sfp}}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}\_\{\backslash text\{sfp\}\}</annotation></semantics>$ has finite coproducts and we can identity $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$ with the category of finite product-preserving functors $<semantics>{C}_{\text{sfp}}^{\text{op}}\to \mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}^\{\backslash text\{op\}\}\_\{\backslash text\{sfp\}\}\; \backslash to\; \backslash mathbf\{Set\}</annotation></semantics>$. One of the most important results of *Notions of Lawvere Theory*, was in expanding the theory to encompass sifted (weighted) colimits. More on this later.

We can play this game with any class of limits $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$. Before defining $<semantics>\mathrm{Phi}<annotation\; encoding="application/x-tex">Phi</annotation></semantics>$-presentability, here is a bit of jargon.

**Definition.** A functor is *$<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-flat* if its colimit commutes with $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-limits.

We call an object $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ of a category $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$ *$<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-presentable* if $<semantics>C(x,-):C\to \mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}(x,-)\; \backslash colon\; \backslash mathbf\{C\}\; \backslash to\; \backslash mathbf\{Set\}</annotation></semantics>$ preserves $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-flat colimits. Given the full subcategory $<semantics>{C}_{\Phi}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}\_\backslash Phi</annotation></semantics>$ of $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-presentable objects, we call $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$ *locally $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-presentable* if it is cocomplete, $<semantics>{C}_{\Phi}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}\_\{\backslash Phi\}</annotation></semantics>$ is small, and any $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$-object is a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-flat colimit of a diagram in $<semantics>{C}_{\Phi}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}\_\{\backslash Phi\}</annotation></semantics>$. Fortunately, we retain the characterization of $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$ being locally $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-presentable if and only if $<semantics>{C}_{\Phi}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}\_\{\backslash Phi\}</annotation></semantics>$ has $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-colimits and $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$ is equivalent to the category $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}({C}_{\Phi}^{\text{op}},\mathrm{Set})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathbf\{C\}^\{\backslash text\{op\}\}\_\backslash Phi,\; \backslash mathbf\{Set\})</annotation></semantics>$ of $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-continuous functors $<semantics>{C}_{\Phi}^{\text{op}}\to \mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}^\{\backslash text\{op\}\}\_\backslash Phi\; \backslash to\; \backslash mathbf\{Set\}</annotation></semantics>$. Important results in *Notions of Lawvere Theory* use the assumption of $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-presentability.

Let’s come back to Lawvere theories. From this point on, we fix a symmetric monoidal closed category $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$ that is both complete and cocomplete. Also, $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ will refer to a class of weights over $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$. Our first task will be to determine what class of limits can replace finite products in the classical case. To this end, we take the following assumption.

**Axiom A.** $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-continuous weights are $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-flat.

This axiom is an analogy with how filtered colimits commute with finite limits in $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$. But for what classes of limits $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ does this hold?

To answer this question, we fix a sound doctrine $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$. Very roughly, a sound doctrine is a collection of small categories whose limits behave nicely with respect to certain colimits. After putting some small assumptions on the underlying category $<semantics>{\mathcal{V}}_{0}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}\_0</annotation></semantics>$ which we’ll sweep under the rug, define $<semantics>{\mathcal{V}}_{\mathbb{D}}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}\_\backslash mathbb\{D\}</annotation></semantics>$ to be the full sub $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-category consisting of those objects $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ such that $<semantics>[x,-]:\mathcal{V}\to \mathcal{V}<annotation\; encoding="application/x-tex">[x,-]\; \backslash colon\; \backslash mathcal\{V\}\; \backslash to\; \backslash mathcal\{V\}</annotation></semantics>$ preserves $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$-flat colimits. Let $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ be the class of limits ‘built from’ conical $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$-limits and $<semantics>{\mathcal{V}}_{\mathbb{D}}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}\_\backslash mathbb\{D\}</annotation></semantics>$-powers in the sense that we take $<semantics>\varphi \in \Phi <annotation\; encoding="application/x-tex">\backslash phi\; \backslash in\; \backslash Phi</annotation></semantics>$ if

any $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-category with conical $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$ limits and $<semantics>{\mathcal{V}}_{\mathbb{D}}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}\_\backslash mathbb\{D\}</annotation></semantics>$-powers also admits $<semantics>\varphi <annotation\; encoding="application/x-tex">\backslash phi</annotation></semantics>$-weighted limits, and

any $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-functor conical $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$ limits and $<semantics>{\mathcal{V}}_{\mathbb{D}}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}\_\backslash mathbb\{D\}</annotation></semantics>$-powers also preserves $<semantics>\varphi <annotation\; encoding="application/x-tex">\backslash phi</annotation></semantics>$-weighted limits.

The fancy way of saying this is that $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ is the *saturation class* of conical $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$-limits and $<semantics>{\mathcal{V}}_{\mathbb{D}}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}\_\backslash mathbb\{D\}</annotation></semantics>$-powers. It’s easy enough to see that $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ contains the conical $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$-limits and $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-powers.

Having constructed a class of limits $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ from a sound doctrine $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$, we use the following theorem to imply that $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ satisfies the axiom above.

**Theorem.** Let $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$ be a small $<semantics>V<annotation\; encoding="application/x-tex">\backslash mathbf\{V\}</annotation></semantics>$-category with $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-weighted limits and $<semantics>F:\mathcal{K}\to \mathcal{V}<annotation\; encoding="application/x-tex">F\; \backslash colon\; \backslash mathcal\{K\}\; \backslash to\; \backslash mathcal\{V\}</annotation></semantics>$ be a $<semantics>V<annotation\; encoding="application/x-tex">\backslash mathbf\{V\}</annotation></semantics>$-functor. The following are equivalent:

$<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ is $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$-continuous;

$<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ is $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-flat

$<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ is $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-continuous.

In particular, the first item allows us to construct $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ using sound limits and the equivalence between the second and third item is precisely the axiom of interest. Here are some examples.

**Example.** Let $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$ be the collection of all finite categories. We will also take $<semantics>{\mathcal{V}}_{0}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}\_0</annotation></semantics>$ be locally finitely presentable with the additional requirement that the monoidal unit $<semantics>I<annotation\; encoding="application/x-tex">I</annotation></semantics>$ is finitely presentable as is the tensor product of two finitely presentable objects. Examples of such a $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$ are categories of sets, abelian groups, modules over a commutative ring, chain complexes, categories, groupoids, and simplicial sets. Then $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$, as constructed from $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$ above gives a good notion of $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-enriched finite limits.

**Example.** A second example, and one of the main contributions of *Notions of Lawvere Theory* is when $<semantics>\mathbb{D}<annotation\; encoding="application/x-tex">\backslash mathbb\{D\}</annotation></semantics>$ is the class of all finite, discrete categories. Here, we take our $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$ as in the first example, though we do not require the monoidal unit to be strongly finitely presentable. We do this because, by requiring the monoidal unit to be strongly finitely presentable, we lose the example where $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$ is the category of directed graphs, which happens to be a key example, particularly to realizing categories as an model of a Lawvere theory. In this case, the induced class $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ gives an enriched version of *strongly finite limits* as discussed above. This $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ generalizes finite products in the sense that they coincide when $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$ is $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$.

## Correspondence between Lawvere theories and monads

Now that we’ve gotten our hands on some suitable limits, let’s see how we can obtain the classical correspondence between Lawvere theories and monads. Naturally, we’ll be assuming axiom A. In addition, we fix a $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-category $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$ that satisfies the following.

**Axiom B1.** $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$ is locally $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-presentable.

This axiom implies, as in our discussions above, that $<semantics>\mathcal{K}\cong \Phi <annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\; \backslash cong\; \backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}({\mathcal{K}}_{\Phi}^{\text{op}},\mathcal{V})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{K\}^\{\backslash text\{op\}\}\_\backslash Phi,\backslash mathcal\{V\})</annotation></semantics>$. This is not particularly restrictive, as presheaf $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-categories are locally $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-presentable.

Now, define a *Lawvere $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory on $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$* to be a bijective-on-objects $<semantics>V<annotation\; encoding="application/x-tex">\backslash mathbf\{V\}</annotation></semantics>$-functor $<semantics>g:{\mathcal{K}}_{\Phi}^{\text{op}}\to \mathcal{L}<annotation\; encoding="application/x-tex">g\backslash colon\; \backslash mathcal\{K\}^\{\backslash text\{op\}\}\_\{\backslash Phi\}\; \backslash to\; \backslash mathcal\{L\}</annotation></semantics>$ that preserves $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-limits. A striking difference between a Lawvere $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory and the classical notion is that the former does not require $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ to have the limits under consideration. This makes defining the models of a Lawvere $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory a subtler issue than in the classical case. Instead of defining a model to be a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-continuous functor as one might expect, we instead use the pullback square

To understand what a model looks like, use the intuition for a pullback in the category $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$ and the fact that $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$ is equivalent to $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}({\mathcal{K}}_{\Phi}^{\text{op}},\mathcal{V})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{K\}^\{\backslash text\{op\}\}\_\backslash Phi,\backslash mathcal\{V\})</annotation></semantics>$. So a model will be a $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-functor $<semantics>\mathcal{L}\to \mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}\; \backslash to\; \backslash mathcal\{V\}</annotation></semantics>$ whose restriction along $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ is $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-continuous.

The other major player in this section is the category of $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-flat monads $<semantics>{\mathrm{Mnd}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Mnd\}\_\{\backslash Phi\}(\backslash mathcal\{K\})</annotation></semantics>$. We claim that there is an equivalence between $<semantics>{\mathrm{Law}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Law\}\_\{\backslash Phi\}(\backslash mathcal\{K\})</annotation></semantics>$ and $<semantics>{\mathrm{Mnd}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Mnd\}\_\{\backslash Phi\}(\backslash mathcal\{K\})</annotation></semantics>$. To verify this, we construct a pair of functors between $<semantics>{\mathrm{Law}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Law\}\_\{\backslash Phi\}(\backslash mathcal\{K\})</annotation></semantics>$ and $<semantics>{\mathrm{Mnd}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Mnd\}\_\{\backslash Phi\}(\backslash mathcal\{K\})</annotation></semantics>$. The first under consideration: $<semantics>\mathrm{mnd}:{\mathrm{Law}}_{\Phi}(\mathcal{K})\to {\mathrm{Mnd}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\{mnd\}\; \backslash colon\; \backslash mathbf\{Law\}\_\{\backslash Phi\}(\backslash mathcal\{K\})\; \backslash to\; \backslash mathbf\{Mnd\}\_\{\backslash Phi\}(\backslash mathcal\{K\})</annotation></semantics>$. We define this with the help of the following proposition.

**Proposition.** The functor $<semantics>u<annotation\; encoding="application/x-tex">u</annotation></semantics>$ from the above pullback diagram is monadic via a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-flat monad $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$.

Hence, a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ gives a monadic functor $<semantics>u:\mathrm{Mod}(\mathcal{L})\to \mathcal{K}<annotation\; encoding="application/x-tex">u\; \backslash colon\; \backslash mathbf\{Mod\}(\backslash mathcal\{L\})\; \backslash to\; \backslash mathcal\{K\}</annotation></semantics>$ that yields a monad $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$ on $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$. Moreover, this monad preserves all the limits required to be an object in $<semantics>{\mathrm{Mnd}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Mnd\}\_\backslash Phi\; (\backslash mathcal\{K\})</annotation></semantics>$. So, define $<semantics>\mathrm{mnd}(\mathcal{L})=t<annotation\; encoding="application/x-tex">\{mnd\}\; (\backslash mathcal\{L\})\; =\; t</annotation></semantics>$.

Next, we define a functor $<semantics>\mathrm{th}:{\mathrm{Mnd}}_{\Phi}(\mathcal{K})\to {\mathrm{Law}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\{th\}\; \backslash colon\; \backslash mathbf\{Mnd\}\_\{\backslash Phi\}(\backslash mathcal\{K\})\; \backslash to\; \backslash mathbf\{Law\}\_\{\backslash Phi\}(\backslash mathcal\{K\})</annotation></semantics>$. Consider a monad $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$ in $<semantics>{\mathrm{Mnd}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Mnd\}\_\{\backslash Phi\}(\backslash mathcal\{K\})</annotation></semantics>$. As per usual, $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$ factors through the Eilenberg-Moore category $<semantics>\mathcal{K}\to {\mathcal{K}}^{t}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\; \backslash to\; \backslash mathcal\{K\}^t</annotation></semantics>$ which we precompose with the inclusion $<semantics>{\mathcal{K}}_{\Phi}\to \mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\_\backslash Phi\; \backslash to\; \backslash mathcal\{K\}</annotation></semantics>$ giving $<semantics>f:{\mathcal{K}}_{\Phi}\to {\mathcal{K}}^{m}<annotation\; encoding="application/x-tex">f\; \backslash colon\; \backslash mathcal\{K\}\_\backslash Phi\; \backslash to\; \backslash mathcal\{K\}^m</annotation></semantics>$. Now defining a $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-category $<semantics>\mathcal{G}<annotation\; encoding="application/x-tex">\backslash mathcal\{G\}</annotation></semantics>$ that has objects from $<semantics>{\mathcal{K}}_{\Phi}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\_\backslash Phi</annotation></semantics>$ and $<semantics>\mathcal{G}(x,y)={\mathcal{K}}^{m}(\mathrm{fx},\mathrm{fy})<annotation\; encoding="application/x-tex">\backslash mathcal\{G\}(x,y)\; =\; \backslash mathcal\{K\}^m(fx,fy)</annotation></semantics>$, we factor $<semantics>f<annotation\; encoding="application/x-tex">f</annotation></semantics>$

where $<semantics>\ell <annotation\; encoding="application/x-tex">\backslash ell</annotation></semantics>$ is bijective-on-objects and $<semantics>r<annotation\; encoding="application/x-tex">r</annotation></semantics>$ is full and faithful. This factorization is unique up to unique isomorphism. Define $<semantics>\text{th}(t)={\mathcal{G}}^{\text{op}}<annotation\; encoding="application/x-tex">\backslash text\{th\}\; (t)\; =\; \backslash mathcal\{G\}^\{\backslash text\{op\}\}</annotation></semantics>$.

At this point, we have functors $$<semantics>\text{mnd}:{\mathrm{Law}}_{\Phi}(\mathcal{K})\to {\mathrm{Mnd}}_{\Phi}(\mathcal{K})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\text{th}:{\mathrm{Mnd}}_{\Phi}(\mathcal{K})\to {\mathrm{Law}}_{\Phi}(\mathcal{K})<annotation\; encoding="application/x-tex">\; \backslash text\{mnd\}\; \backslash colon\; \backslash mathbf\{Law\}\_\{\backslash Phi\}(\backslash mathcal\{K\})\; \backslash to\; \backslash mathbf\{Mnd\}\_\{\backslash Phi\}(\backslash mathcal\{K\})\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash text\{th\}\; \backslash colon\; \backslash mathbf\{Mnd\}\_\{\backslash Phi\}(\backslash mathcal\{K\})\; \backslash to\; \backslash mathbf\{Law\}\_\{\backslash Phi\}(\backslash mathcal\{K\})\; </annotation></semantics>$$ so let’s turn our attention to showing that these are mutual weak inverses. The first step is to show that the category of algebras $<semantics>{\mathcal{K}}^{t}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}^t</annotation></semantics>$ for a given monad $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$ is the category of models $<semantics>\mathrm{Mod}(\text{th}(t))<annotation\; encoding="application/x-tex">\backslash mathbf\{Mod\}(\backslash text\{th\}\; (t))</annotation></semantics>$.

**Theorem 6.6.** The $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-functor
$$<semantics>{\mathcal{K}}^{t}(r-,-):{\mathcal{K}}^{t}\to [\text{th}(t{)}^{\text{op}},\mathcal{V}],\phantom{\rule{1em}{0ex}}x\mapsto {\mathcal{K}}^{m}(r-,x)<annotation\; encoding="application/x-tex">\; \backslash mathcal\{K\}^t(r-,-)\; \backslash colon\; \backslash mathcal\{K\}^t\; \backslash to\; [\backslash text\{th\}\; (t)^\{\backslash text\{op\}\},\backslash mathcal\{V\}],\; \backslash quad\; x\; \backslash mapsto\; \backslash mathcal\{K\}^m(r-,x)\; </annotation></semantics>$$
restricts to an isomorphism of $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-categories $<semantics>{\mathcal{K}}^{t}\cong \mathrm{Mod}(\text{th}(t))<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}^t\; \backslash cong\; \backslash mathbf\{Mod\}(\backslash text\{th\}\; (t))</annotation></semantics>$.

This theorem gives us that $<semantics>\text{mnd}\circ \text{th}\cong \text{id}<annotation\; encoding="application/x-tex">\backslash text\{mnd\}\; \backslash circ\; \backslash text\{th\}\; \backslash cong\; \backslash text\{id\}</annotation></semantics>$. The next theorem gives us the other direction.

**Theorem 6.7.** There is an isomorphism $<semantics>\text{th}\circ \text{mnd}\cong \text{id}<annotation\; encoding="application/x-tex">\backslash text\{th\}\; \backslash circ\; \backslash text\{mnd\}\; \backslash cong\; \backslash text\{id\}</annotation></semantics>$.

Let’s sketch the proof. Let $<semantics>g:{\mathcal{K}}_{\Phi}^{\text{op}}\to \mathcal{L}<annotation\; encoding="application/x-tex">g\; \backslash colon\; \backslash mathcal\{K\}^\{\backslash text\{op\}\}\_\{\backslash Phi\}\; \backslash to\; \backslash mathcal\{L\}</annotation></semantics>$ be a Lawvere $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory. If we denote $<semantics>\text{mnd}(\mathcal{T})<annotation\; encoding="application/x-tex">\backslash text\{mnd\}\; (\backslash mathcal\{T\})</annotation></semantics>$ by $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$, we get $<semantics>\text{th}\circ \text{mnd}(\mathcal{T})=\text{th}(t)={\mathcal{G}}^{\text{op}}<annotation\; encoding="application/x-tex">\backslash text\{th\}\; \backslash circ\; \backslash text\{mnd\}\; (\backslash mathcal\{T\})\; =\; \backslash text\{th\}\; (t)\; =\; \backslash mathcal\{G\}^\{\backslash text\{op\}\}</annotation></semantics>$ via the factorization

where $<semantics>\ell <annotation\; encoding="application/x-tex">\backslash ell</annotation></semantics>$ is bijective-on-objects and $<semantics>r<annotation\; encoding="application/x-tex">r</annotation></semantics>$ is fully faithful. It remains to show that $<semantics>\mathcal{T}={\mathcal{G}}^{\text{op}}<annotation\; encoding="application/x-tex">\backslash mathcal\{T\}\; =\; \backslash mathcal\{G\}^\{\backslash text\{op\}\}</annotation></semantics>$.

Let’s compute the image of an $<semantics>{\mathcal{K}}_{\Phi}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\_\backslash Phi</annotation></semantics>$-object $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ in $<semantics>{\mathrm{Mod}}_{\Phi}(\text{th}(t))<annotation\; encoding="application/x-tex">\backslash mathbf\{Mod\}\_\{\backslash Phi\}\; (\backslash text\{th\}\; (t))</annotation></semantics>$. For this, recall that we have $<semantics>\mathcal{K}\simeq \Phi <annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\; \backslash simeq\; \backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}({\mathcal{K}}_{\Phi}^{\text{op}},\mathcal{V})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{K\}^\{\backslash text\{op\}\}\_\{\backslash Phi\},\backslash mathcal\{V\})</annotation></semantics>$ by assumption. Embedding $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ into $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cats}({\mathcal{K}}_{\Phi},\mathcal{V})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cats\}(\backslash mathcal\{K\}\_\{\backslash Phi\},\backslash mathcal\{V\})</annotation></semantics>$ gives us
$$<semantics>{\mathcal{K}}_{\Phi}(-,x):{\mathcal{K}}_{\Phi}^{\text{op}}\to \mathcal{V}.<annotation\; encoding="application/x-tex">\; \backslash mathcal\{K\}\_\backslash Phi(-,x)\; \backslash colon\; \backslash mathcal\{K\}^\{\backslash text\{op\}\}\_\{\backslash Phi\}\; \backslash to\; \backslash mathcal\{V\}.\; </annotation></semantics>$$
This, in turn, is mapped to the left Kan extension
$$<semantics>{\text{Lan}}_{g}({\mathcal{K}}_{\Phi}(-,x)):\mathcal{T}\to \mathcal{V}<annotation\; encoding="application/x-tex">\; \backslash text\{Lan\}\_g\; (\backslash mathcal\{K\}\_\backslash Phi(-,x))\; \backslash colon\; \backslash mathcal\{T\}\; \backslash to\; \backslash mathcal\{V\}\; </annotation></semantics>$$
along $<semantics>g:{\mathcal{K}}_{\Phi}^{\text{op}}\to \mathcal{T}<annotation\; encoding="application/x-tex">g\; \backslash colon\; \backslash mathcal\{K\}^\{\backslash text\{op\}\}\_\{\backslash Phi\}\; \backslash to\; \backslash mathcal\{T\}</annotation></semantics>$ (the Lawvere $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ theory we began with). Here, we can compute that $<semantics>{\text{Lan}}_{g}({\mathcal{K}}_{\Phi}(-,x))<annotation\; encoding="application/x-tex">\backslash text\{Lan\}\_g\; (\backslash mathcal\{K\}\_\backslash Phi(-,x))</annotation></semantics>$ is $<semantics>\mathcal{T}(-,\mathrm{gx})<annotation\; encoding="application/x-tex">\backslash mathcal\{T\}(-,gx)</annotation></semantics>$ meaning the factorization above is

Therefore, $<semantics>\mathcal{T}={\mathcal{G}}^{\text{op}}<annotation\; encoding="application/x-tex">\backslash mathcal\{T\}\; =\; \backslash mathcal\{G\}^\{\backslash text\{op\}\}</annotation></semantics>$ as desired.

## Many-sorted theories

Moving from single-sorted to many-sorted theories, we will take a different assumption on our $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-category $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$.

**Axiom B2.** $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$ is a $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-category with $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-limits and such that the Yoneda inclusion $<semantics>\mathcal{K}\to [{\mathcal{K}}^{\text{op}},\mathcal{V}]<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\; \backslash to\; [\backslash mathcal\{K\}^\{\backslash text\{op\}\}\; ,\; \backslash mathcal\{V\}]</annotation></semantics>$ has a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-continuous left adjoint.

This requirement on $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$ is not overly restrictive as it holds for all presheaf $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-categories and all Grothendieck topoi when $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$ is $<semantics>\mathrm{Set}<annotation\; encoding="application/x-tex">\backslash mathbf\{Set\}</annotation></semantics>$. The nice thing about this assumption is that we can compute all colimits and $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-limits in $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$ by passing to $<semantics>[{\mathcal{K}}^{\text{op}},\mathcal{V}]<annotation\; encoding="application/x-tex">[\backslash mathcal\{K\}^\{\backslash text\{op\}\}\; ,\; \backslash mathcal\{V\}]</annotation></semantics>$, where they commute, then reflecting back.

Generalizing Lawvere theories here is a bit simpler than in the previous section. Indeed, call any small $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-category $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ with $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-limits a *$<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory*. Notice that we no longer have a bijective-on-objects functor involved in the definition. That functor forced the single-sortedness. With the functor no longer constraining the structure, we have the possibility for many sorts. Also, a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory does have all $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-limits here, unlike in the single-sorted case. This allows for a much simpler definition of a model. Indeed, the *category of models* for a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ is the full subcategory $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathcal{L},\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{L\},\; \backslash mathcal\{K\})</annotation></semantics>$ of $<semantics>[\mathcal{L},\mathcal{K}]<annotation\; encoding="application/x-tex">[\backslash mathcal\{L\},\; \backslash mathcal\{K\}]</annotation></semantics>$.

Presently, we are interested in generalizing two important properties of Lawvere theory to $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theories. The first is that algebraic functors have left adjoints. The second is the reflectiveness of models.

**Algebraic functors have left adjoints.** A *morphism of $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theories* is a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-continuous $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-functor $<semantics>g:\mathcal{L}\to \mathcal{L}{\textstyle \prime}<annotation\; encoding="application/x-tex">g\; \backslash colon\; \backslash mathcal\{L\}\; \backslash to\; \backslash mathcal\{L\}\backslash prime</annotation></semantics>$. Any such morphism induces a pullback $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-functor $<semantics>{g}^{*}:\Phi <annotation\; encoding="application/x-tex">g^\{\backslash ast\}\; \backslash colon\; \backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathcal{L}{\textstyle \prime},\mathcal{K})\to \Phi <annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{L\}\backslash prime,\; \backslash mathcal\{K\})\; \backslash to\; \backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathcal{L},\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{L\},\; \backslash mathcal\{K\})</annotation></semantics>$ between model $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-categories. We call such functors *$<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-algebraic*. And yes, these do have left adjoints just as in the context of classical Lawvere theories.

**Theorem.** Let $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ and $<semantics>\mathcal{L}{\textstyle \prime}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}\backslash prime</annotation></semantics>$ be $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theories and $<semantics>g:\mathcal{L}\to \mathcal{L}{\textstyle \prime}<annotation\; encoding="application/x-tex">g\; \backslash colon\; \backslash mathcal\{L\}\; \backslash to\; \backslash mathcal\{L\}\backslash prime</annotation></semantics>$ a $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-functor between them. Given a model $<semantics>m:\mathcal{L}\to \mathcal{K}<annotation\; encoding="application/x-tex">m\; \backslash colon\; \backslash mathcal\{L\}\; \backslash to\; \backslash mathcal\{K\}</annotation></semantics>$, then $<semantics>{\text{Lan}}_{g}m:\mathcal{L}{\textstyle \prime}\to \mathcal{K}<annotation\; encoding="application/x-tex">\backslash text\{Lan\}\_g\; m\; \backslash colon\; \backslash mathcal\{L\}\backslash prime\; \backslash to\; \backslash mathcal\{K\}</annotation></semantics>$ is a model.

What is happening here? Of course, pulling back by $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ gives a way to turn models of $<semantics>\mathcal{L}{\textstyle \prime}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}\backslash prime</annotation></semantics>$ into models of $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ – this is the algebraic functor $<semantics>{g}^{*}<annotation\; encoding="application/x-tex">g^\{\backslash ast\}</annotation></semantics>$. But the left Kan extension along $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ gives a way to turn a model $<semantics>m<annotation\; encoding="application/x-tex">m</annotation></semantics>$ of $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ into a model of $<semantics>\mathcal{L}{\textstyle \prime}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}\backslash prime</annotation></semantics>$ as depicted in the diagram

This theorem says that process gives a functor $<semantics>{g}_{*}:\Phi <annotation\; encoding="application/x-tex">g\_\{\backslash ast\}\; \backslash colon\; \backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathcal{L},\mathcal{K})\to \Phi <annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{L\},\; \backslash mathcal\{K\})\; \backslash to\; \backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathcal{L}{\textstyle \prime},\mathcal{K})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{L\}\backslash prime,\; \backslash mathcal\{K\})</annotation></semantics>$ given by $<semantics>m\mapsto {\text{Lan}}_{g}m<annotation\; encoding="application/x-tex">m\; \backslash mapsto\; \backslash text\{Lan\}\_g\; m</annotation></semantics>$.

We can prove this theorem for $<semantics>\mathcal{K}=\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\; =\; \backslash mathcal\{V\}</annotation></semantics>$ without requiring axiom B2. This axiom is used to extend this result to a $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-category $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$. The existence of the left adjoint $<semantics>\ell <annotation\; encoding="application/x-tex">\backslash ell</annotation></semantics>$ to the Yoneda embedding $<semantics>y<annotation\; encoding="application/x-tex">y</annotation></semantics>$ of $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$ gives a factorization $<semantics>{\text{Lan}}_{g}m=\ell {\text{Lan}}_{g}ym<annotation\; encoding="application/x-tex">\backslash text\{Lan\}\_g\; m\; =\; \backslash ell\; \backslash text\{Lan\}\_g\; y\; m</annotation></semantics>$. The proof then reduces to showing that $<semantics>{\text{Lan}}_{g}ym<annotation\; encoding="application/x-tex">\backslash text\{Lan\}\_g\; y\; m</annotation></semantics>$ is $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-continuous, since we are already assuming that $<semantics>\ell <annotation\; encoding="application/x-tex">\backslash ell</annotation></semantics>$ is. But because the codomain of $<semantics>{\text{Lan}}_{g}ym<annotation\; encoding="application/x-tex">\backslash text\{Lan\}\_g\; y\; m</annotation></semantics>$ is $<semantics>[{\mathcal{K}}^{\text{op}},\mathcal{V}]<annotation\; encoding="application/x-tex">[\backslash mathcal\{K\}^\{\backslash text\{op\}\},\backslash mathcal\{V\}]</annotation></semantics>$, we can rest on the fact that we have proven the result for $<semantics>\mathcal{K}=\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\; =\; \backslash mathcal\{V\}</annotation></semantics>$. Limits are taken pointwise, after all. Actually, the left adjoint to $<semantics>{g}^{*}<annotation\; encoding="application/x-tex">g^\{\backslash ast\}</annotation></semantics>$ holds more generally, but our assumptions on $<semantics>\mathcal{K}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}</annotation></semantics>$ allow us to explicitly compute the left adjoint with left Kan extensions.

**Reflexiveness of models.** Having discussed left adjoints of algebraic functors, we now move on to show that categories of models $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathcal{T},\mathcal{V})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{T\},\backslash mathcal\{V\})</annotation></semantics>$ are reflexive in $<semantics>[\mathcal{T},\mathcal{V}]<annotation\; encoding="application/x-tex">[\backslash mathcal\{T\},\; \backslash mathcal\{V\}]</annotation></semantics>$. Consider the free-forgetful (ordinary) adjunction

between $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-categories and those $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-categories with $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-limits and functors preserving them. Given a $<semantics>\mathcal{V}<annotation\; encoding="application/x-tex">\backslash mathcal\{V\}</annotation></semantics>$-category $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ in the image of $<semantics>U<annotation\; encoding="application/x-tex">U</annotation></semantics>$. Note that $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ is a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory. It follows from this adjunction that $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathrm{\mathcal{F}\mathcal{L}},\mathcal{V})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{FL\},\backslash mathcal\{V\})</annotation></semantics>$ is equivalent to the category $<semantics>[\mathcal{L},\mathcal{V}]<annotation\; encoding="application/x-tex">[\backslash mathcal\{L\},\backslash mathcal\{V\}]</annotation></semantics>$. Moreover, since $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ has $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-limits, the inclusion $<semantics>\mathcal{L}\hookrightarrow \mathrm{\mathcal{F}\mathcal{L}}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}\; \backslash hookrightarrow\; \backslash mathcal\{FL\}</annotation></semantics>$ has a right adjoint $<semantics>R<annotation\; encoding="application/x-tex">R</annotation></semantics>$ inducing an algebraic functor $<semantics>{R}^{*}:\Phi <annotation\; encoding="application/x-tex">R^\{\backslash ast\}\; \backslash colon\; \backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathcal{L},\mathcal{V})\to \Phi <annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{L\},\backslash mathcal\{V\})\; \backslash to\; \backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathrm{\mathcal{F}\mathcal{L}},\mathcal{V})\simeq [\mathcal{L},\mathcal{V}]<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{FL\},\backslash mathcal\{V\})\; \backslash simeq\; [\backslash mathcal\{L\},\backslash mathcal\{V\}]</annotation></semantics>$. But we just showed that algebraic functors have left adjoints, giving us the following theorem.

**Theorem.** $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-$<semantics>\mathrm{Cts}(\mathcal{L},\mathcal{V})<annotation\; encoding="application/x-tex">\backslash mathbf\{Cts\}(\backslash mathcal\{L\},\backslash mathcal\{V\})</annotation></semantics>$ is reflective in $<semantics>[\mathcal{L},\mathcal{V}]<annotation\; encoding="application/x-tex">[\backslash mathcal\{L\},\backslash mathcal\{V\}]</annotation></semantics>$.

As promised, in the two general contexts corresponding to the axioms B1 and B2, we have the Lawvere theory-monad correspondence, that algebraic functors have left adjoints, and that categories of models are reflective.

## An example

After all of that abstract nonsense, let’s get our feet back on the ground. Here is an example of manifesting a category with a chosen terminal object as a generalized Lawvere theory. This comes courtesy of Nishizawa and Power.

Let $<semantics>0<annotation\; encoding="application/x-tex">\backslash mathbf\{0\}</annotation></semantics>$ denote the empty category, $<semantics>1<annotation\; encoding="application/x-tex">\backslash mathbf\{1\}</annotation></semantics>$ the terminal category, and $<semantics>2<annotation\; encoding="application/x-tex">\backslash mathbf\{2\}</annotation></semantics>$ the category $<semantics>\{a\to b\}<annotation\; encoding="application/x-tex">\backslash \{a\; \backslash to\; b\backslash \}</annotation></semantics>$ with two objects and a single arrow between them. We will also take $<semantics>\mathcal{K}=\mathcal{V}=\mathrm{Cat}<annotation\; encoding="application/x-tex">\backslash mathcal\{K\}\; =\; \backslash mathcal\{V\}\; =\; \backslash mathbf\{Cat\}</annotation></semantics>$. The class of limits $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$ here is the finite $<semantics>\mathrm{Cat}<annotation\; encoding="application/x-tex">\backslash mathbf\{Cat\}</annotation></semantics>$-powers. We define a Lawvere $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ to be the $<semantics>\mathrm{Cat}<annotation\; encoding="application/x-tex">\backslash mathbf\{Cat\}</annotation></semantics>$-category where we formally add to $<semantics>{\mathrm{Cat}}_{\text{fp}}^{\text{op}}<annotation\; encoding="application/x-tex">\backslash mathbf\{Cat\}^\{\backslash text\{op\}\}\_\{\backslash text\{fp\}\}</annotation></semantics>$ (the opposite full subcategory on the finitely presentable objects) two arrows: $<semantics>\tau :0\to 1<annotation\; encoding="application/x-tex">\backslash tau\; \backslash colon\; \backslash mathbf\{0\}\; \backslash to\; \backslash mathbf\{1\}</annotation></semantics>$ and $<semantics>\sigma :1\to 2<annotation\; encoding="application/x-tex">\backslash sigma\; \backslash colon\; \backslash mathbf\{1\}\; \backslash to\; \backslash mathbf\{2\}</annotation></semantics>$. We then close up under finite $<semantics>\mathrm{Cat}<annotation\; encoding="application/x-tex">\backslash mathbf\{Cat\}</annotation></semantics>$-powers and modulo the commutative diagrams

$<semantics>\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}<annotation\; encoding="application/x-tex">\backslash quad\; \backslash quad</annotation></semantics>$ $<semantics>\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}<annotation\; encoding="application/x-tex">\backslash quad\; \backslash quad</annotation></semantics>$

Now $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ is the Lawvere $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-theory for a category with a chosen terminal object. A model of $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$ is a $<semantics>\Phi <annotation\; encoding="application/x-tex">\backslash Phi</annotation></semantics>$-continuous $<semantics>\mathrm{Cat}<annotation\; encoding="application/x-tex">\backslash mathbf\{Cat\}</annotation></semantics>$-functor $<semantics>\mathcal{L}\to \mathrm{Cat}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}\; \backslash to\; \backslash mathbf\{Cat\}</annotation></semantics>$. This means that if $<semantics>M<annotation\; encoding="application/x-tex">M</annotation></semantics>$ is a model, it must preserve powers and so the following diagrams must commute:

$<semantics>\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}<annotation\; encoding="application/x-tex">\backslash quad\; \backslash quad</annotation></semantics>$ $<semantics>\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}<annotation\; encoding="application/x-tex">\backslash quad\; \backslash quad</annotation></semantics>$

Here $<semantics>\text{dom}<annotation\; encoding="application/x-tex">\backslash text\{dom\}</annotation></semantics>$ and $<semantics>\text{cod}<annotation\; encoding="application/x-tex">\backslash text\{cod\}</annotation></semantics>$ choose the domain and codomain, and $<semantics>\Delta <annotation\; encoding="application/x-tex">\backslash Delta</annotation></semantics>$ the diagonal functor. Note that the commutativity of these diagrams witnesses the preservation of raising the terminal category to the first three diagrams given. Let’s parse these diagrams out.

The category we get from the model $<semantics>M<annotation\; encoding="application/x-tex">M</annotation></semantics>$ is $<semantics>M1<annotation\; encoding="application/x-tex">M1</annotation></semantics>$ and the distinguished terminal object $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$ is chosen by $<semantics>M\tau <annotation\; encoding="application/x-tex">M\backslash tau</annotation></semantics>$. The first two diagrams provide a morphism $<semantics>x\to t<annotation\; encoding="application/x-tex">x\; \backslash to\; t</annotation></semantics>$ for every object $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ in $<semantics>M1<annotation\; encoding="application/x-tex">M1</annotation></semantics>$. The third diagram gives the identity map on $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$. The uniqueness of maps into $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$ follows from the functorality of $<semantics>M\sigma <annotation\; encoding="application/x-tex">M\backslash sigma</annotation></semantics>$ and $<semantics>\text{cod}<annotation\; encoding="application/x-tex">\backslash text\{cod\}</annotation></semantics>$.

Conversely, given a category $<semantics>C<annotation\; encoding="application/x-tex">\backslash mathbf\{C\}</annotation></semantics>$ with a chosen terminal object $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$, define a model $<semantics>M:\mathcal{L}\to \mathrm{Cat}<annotation\; encoding="application/x-tex">M\; \backslash colon\; \backslash mathcal\{L\}\; \backslash to\; \backslash mathbf\{Cat\}</annotation></semantics>$ by $<semantics>1\to C<annotation\; encoding="application/x-tex">\backslash mathbf\{1\}\; \backslash to\; \backslash mathbf\{C\}</annotation></semantics>$ and $<semantics>{1}^{x}\to {C}^{x}<annotation\; encoding="application/x-tex">\backslash mathbf\{1\}^x\; \backslash to\; \backslash mathbf\{C\}^x</annotation></semantics>$ for $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$-objects $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$. Also, let $<semantics>M\tau <annotation\; encoding="application/x-tex">M\backslash tau</annotation></semantics>$ choose $<semantics>t<annotation\; encoding="application/x-tex">t</annotation></semantics>$ and let $<semantics>M\sigma <annotation\; encoding="application/x-tex">M\; \backslash sigma</annotation></semantics>$ send each $<semantics>\mathcal{L}<annotation\; encoding="application/x-tex">\backslash mathcal\{L\}</annotation></semantics>$-object $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ to the $<semantics>!:x\to t<annotation\; encoding="application/x-tex">!\; \backslash colon\; x\; \backslash to\; t</annotation></semantics>$.

## May 05, 2017

### John Baez - Azimuth

I think of sulfur and phosphorus as clever chameleons of the periodic table: both come in many different forms, called allotropes. There’s white phosphorus, red phosphorus, violet phosphorus and black phosphorus:

and there are about two dozen allotropes of sulfur, with a phase diagram like this:

So I should have guessed that sulfur and phosphorus combine to make many different compounds. But I never thought about this until yesterday!

I’m a great fan of diamonds, not for their monetary value but for the math of their crystal structure:

In a diamond the carbon atoms do not form a lattice in the strict mathematical sense (which is more restrictive than the sense of this word in crystallography). The reason is that there aren’t translational symmetries carrying any atom to any other. Instead, there are two lattices of atoms, shown as red and blue in this picture by Greg Egan. Each atom has 4 nearest neighbors arranged at the vertices of a regular tetrahedron; the tetrahedra centered at the blue atoms are ‘right-side up’, while those centered at the red atoms are ‘upside down’.

Having thought about this a lot, I was happy to read about adamantane. It’s a compound with 10 carbons and 16 hydrogens. There are 4 carbons at the vertices of a regular tetrahedron, and 6 along the edges—but the edges bend out in such a way that the carbons form a tiny piece of a diamond crystal:

or more abstractly, focusing on the carbons and their bonds:

Yesterday I learned that phosphorus decasulfide, P_{4}S_{10}, follows the same pattern:

The angles deviate slightly from the value of

that we’d have in a fragment of a mathematically ideal diamond crystal, but that’s to be expected.

It turns out there are lots of other phosphorus sulfides! Here are some of them:

**Puzzle 1.** Why do each of these compounds have exactly 4 phosphorus atoms?

I don’t know the answer! I can’t believe it’s impossible to form phosphorus–sulfur compounds with some other number of phosphorus atoms, but the Wikipedia article containing this chart says

All known molecular phosphorus sulfides contain a tetrahedral array of four phosphorus atoms. P

_{4}S_{2}is also known but is unstable above −30 °C.

All these phosphorus sulfides contain at most 10 sulfur atoms. If we remove one sulfur from phosphorus decasulfide we can get this:

This is the ‘alpha form’ of P_{4}S_{9}. There’s also a beta form, shown in the chart above.

Some of the phosphorus sulfides have pleasing symmetries, like the

alpha form of P_{4}S_{4}:

or the epsilon form of P_{4}S_{6}:

Others look awkward. The alpha form of P_{4}S_{5} is an ungainly beast:

They all seem to have a few things in common:

• There are 4 phosphorus atoms.

• Each phosphorus atom is connected to 3 or 4 atoms, at most one of which is phosphorus.

• Each sulfur atom is connected to 1 or 2 atoms, which must all be phosphorus.

The pictures seem pretty consistent about showing a ‘double bond’ when a sulfur atom is connected to just 1 phosphorus. However, they don’t show a double bond when a phosphorus atom is connected to just 3 sulfurs.

**Puzzle 2.** Can you draw molecules obeying the 3 rules listed above that aren’t on the chart?

Of all the phosphorus sulfides, P_{4}S_{10} is not only the biggest and most symmetrical, it’s also the most widely used. Humans make thousands of tons of the stuff! It’s used for producing organic sulfur compounds.

People also make P_{4}S_{3}: it’s used in strike-anywhere matches. This molecule is not on the chart I showed you, and it also violates one of the rules I made up:

Somewhat confusingly, P_{4}S_{10} is not only called phosphorus decasulfide: it’s also called phosphorus pentasulfide. Similarly, P_{4}S_{3} is called phosphorus sesquisulfide. Since the prefix ‘sesqui-’ means ‘one and a half’, there seems to be some kind of division by 2 going on here.

## May 04, 2017

### Symmetrybreaking - Fermilab/SLAC

A new result from the Daya Bay collaboration reveals both limitations and strengths of experiments studying antineutrinos at nuclear reactors.

As nuclear reactors burn through fuel, they produce a steady flow of particles called neutrinos. Neutrinos interact so rarely with other matter that they can flow past the steel and concrete of a power plant’s containment structures and keep on moving through anything else that gets in their way.

Physicists interested in studying these wandering particles have taken advantage of this fact by installing neutrino detectors nearby. A recent result using some of these detectors demonstrated both their limitations and strengths.

### The reactor antineutrino anomaly

In 2011, a group of theorists noticed that several reactor-based neutrino experiments had been publishing the same, surprising result: They weren’t detecting as many neutrinos as they thought they would.

Or rather, to be technically correct, they weren’t seeing as many *anti*neutrinos as they thought they would; nuclear reactors actually produce the antimatter partners of the elusive particles. About 6 percent of the expected antineutrinos just weren’t showing up. They called it “the reactor antineutrino anomaly.”

The case of the missing neutrinos was a familiar one. In the 1960s, the Davis experiment located in Homestake Mine in South Dakota reported a shortage of neutrinos coming from processes in the sun. Other experiments confirmed the finding. In 2001, the Sudbury Neutrino Observatory in Ontario demonstrated that the missing neutrinos weren’t missing at all; they had only undergone a bit of a costume change.

Neutrinos come in three types. Scientists discovered that neutrinos could transform from one type to another. The missing neutrinos had changed into a different type of neutrino that the Davis experiment couldn’t detect.

Since 2011, scientists have wondered whether the reactor antineutrino anomaly was a sign of an undiscovered type of neutrino, one that was even harder to detect, called a sterile neutrino.

A new result from the Daya Bay experiment in China not only casts doubt on that theory, it also casts doubt on the idea that scientists understand their model of reactor processes well enough at this time to use it to search for sterile neutrinos.

### The word from Daya Bay

The Daya Bay experiment studies antineutrinos coming from six nuclear reactors on the southern coast of China, about 35 miles northeast of Hong Kong. The reactors are powered by the fission of uranium. Over time, the amount of uranium inside the reactor decreases while the amount of plutonium increases. The fuel is changed—or cycled—about every 18 months.

The main goal of the Daya Bay experiment was to look for the rarest of the known neutrino oscillations. It did that, making a groundbreaking discovery after just nine weeks of data-taking.

But that wasn’t the only goal of the experiment. “We realized right from the beginning that it is important for Daya Bay to address as many interesting physics problems as possible,” says Daya Bay co-spokesperson Kam-Biu Luk of the University of California, Berkeley and the US Department of Energy’s Lawrence Berkeley National Laboratory.

For this result, Daya Bay scientists took advantage of their enormous collection of antineutrino data to expand their investigation to the reactor antineutrino anomaly.

Using data from more than 2 million antineutrino interactions and information about when the power plants refreshed the uranium in each reactor, Daya Bay physicists compared the measurements of antineutrinos coming from different parts of the fuel cycle: early ones dominated by uranium through later ones dominated by both uranium and plutonium.

In theory, the type of fuel producing the antineutrinos should not affect the rate at which they transform into sterile neutrinos. According to Bob Svoboda, chair of the Department of Physics at the University of California, Davis, “a neutrino wouldn’t care how it got made.” But Daya Bay scientists found that the shortage of antineutrinos existed only in processes dominated by uranium.

Their conclusion is that, once again, the missing neutrinos aren’t actually missing. This time, the problem of the missing antineutrinos seems to stem from our understanding of how uranium burns in nuclear power plants. The predictions for how many antineutrinos the scientists should detect may have been overestimated.

“Most of the problem appears to come from the uranium-235 model (uranium-235 is a fissile isotope of uranium), not from the neutrinos themselves,” Svoboda says. “We don’t fully understand uranium, so we have to take any anomaly we measured with a grain of salt.”

This knock against the reactor antineutrino anomaly does not disprove the existence of sterile neutrinos. Other, non-reactor experiments have seen different possible signs of their influence. But it does put a damper on the only evidence of sterile neutrinos to have come from reactor experiments so far.

Other reactor neutrino experiments, such as NEOS in South Korea and PROSPECT in the United States will fill in some missing details. NEOS scientists directly measured antineutrinos coming from reactors in the Hanbit nuclear power complex using a detector placed about 80 feet away, a distance some scientists believe is optimal for detecting sterile neutrinos should they exist. PROSPECT scientists will make the first precision measurement of antineutrinos coming from a highly enriched uranium core, one that does not produce plutonium as it burns.

### A silver lining

The Daya Bay result offers the most detailed demonstration yet of scientists’ ability to use neutrino detectors to peer inside running nuclear reactors.

“As a study of reactors, this is a tour de force,” says theorist Alexander Friedland of SLAC National Accelerator Laboratory. “This is an explicit demonstration that the composition of the reactor fuel has an impact on the neutrinos.”

Some scientists are interested in monitoring nuclear power plants to find out if nuclear fuel is being diverted to build nuclear weapons.

“Suppose I declare my reactor produces 100 kilograms of plutonium per year,” says Adam Bernstein of Lawrence Livermore National Laboratory. “Then I operate it in a slightly different way, and at the end of the year I have 120 kilograms.” That 20-kilogram surplus, left unmeasured, could potentially be moved into a weapons program.

Current monitoring techniques involve checking what goes into a nuclear power plant before the fuel cycle begins and then checking what comes out after it ends. In the meantime, what happens inside is a mystery.

Neutrino detectors allow scientists to understand what’s going on in a nuclear reactor in real time.

Scientists have known for decades that neutrino detectors could be useful for nuclear nonproliferation purposes. Scientists studying neutrinos at the Rovno Nuclear Power Plant in Ukraine first demonstrated that neutrino detectors could differentiate between uranium and plutonium fuel.

Most of the experiments have done this by looking at changes in the aggregate number of antineutrinos coming from a detector. Daya Bay showed that neutrino detectors could track the plutonium inventory in nuclear fuel by studying the energy spectrum of antineutrinos produced.

“The most likely use of neutrino detectors in the near future is in so-called ‘cooperative agreements,’ where a $20-million-scale neutrino detector is installed in the vicinity of a reactor site as part of a treaty,” Svoboda says. “The site can be monitored very reliably without having to make intrusive inspections that bring up issues of national sovereignty.”

Luk says he is dubious that the idea will take off, but he agrees that Daya Bay has shown that neutrino detectors can give an incredibly precise report. “This result is the best demonstration so far of using a neutrino detector to probe the heartbeat of a nuclear reactor.”

## May 03, 2017

### ZapperZ - Physics and Physicists

The Office of Science supports six research programs, and there were winners and losers among them. On the plus side, advanced scientific computing research, which funds much of DOE's supercomputing capabilities, gets a 4.2% increase to $647 million. High energy physics gets a boost of 3.8% to $825 million. Basic energy sciences, which funds work in chemistry, material science, and condensed matter physics and runs most of DOE's large user facilities, gets a bump up of 1.2% to $1.872 billion. Nuclear physics gets a 0.8% raise to $622 million; biological and environmental research inches up 0.5% to $612 million. In contrast, the fusion energy sciences program sees its budget fall a whopping 13.2% to $380 million.

It will continue to be challenging for physics funding during the next foreseeable future, but at least this will not cause a major panic. I've been highly critical of the US Congress on many issues, but I will tip my hat to them this time for standing up to the ridiculous budget that came out of the Trump administration earlier.

Zz.

## May 02, 2017

### Symmetrybreaking - Fermilab/SLAC

According to the Fermi LAT collaboration, the galaxy’s excessive gamma-ray glow likely comes from pulsars, the remains of collapsed ancient stars.

A mysterious gamma-ray glow at the center of the Milky Way is most likely caused by pulsars, the incredibly dense, rapidly spinning cores of collapsed ancient stars that were up to 30 times more massive than the sun.

That’s the conclusion of a new analysis by an international team of astrophysicists on the Fermi LAT collaboration. The findings cast doubt on previous interpretations of the signal as a potential sign of dark matter, a form of matter that accounts for 85 percent of all matter in the universe but that so far has evaded detection.

“Our study shows that we don’t need dark matter to understand the gamma-ray emissions of our galaxy,” says Mattia Di Mauro from the Kavli Institute for Particle Astrophysics and Cosmology, a joint institute of Stanford University and the US Department of Energy's SLAC National Accelerator Laboratory. “Instead, we have identified a population of pulsars in the region around the galactic center, which sheds new light on the formation history of the Milky Way.”

Di Mauro led the analysis, which looked at the glow with the Large Area Telescope on NASA’s Fermi Gamma-ray Space Telescope, which has been orbiting Earth since 2008. The LAT, a sensitive “eye” for gamma rays, the most energetic form of light, was conceived of and assembled at SLAC, which also hosts its operations center.

The collaboration’s findings, submitted to *The Astrophysical Journal* for publication, are available as a preprint.

### A mysterious glow

Dark matter is one of the biggest mysteries of modern physics. Researchers know that dark matter exists because it bends light from distant galaxies and affects how galaxies rotate. But they don’t know what the substance is made of. Most scientists believe it’s composed of yet-to-be-discovered particles that almost never interact with regular matter other than through gravity, making it very hard to detect them.

One way scientific instruments might catch a glimpse of dark matter particles is when the particles either decay or collide and destroy each other. “Widely studied theories predict that these processes would produce gamma rays,” says Seth Digel, head of KIPAC’s Fermi group. “We search for this radiation with the LAT in regions of the universe that are rich in dark matter, such as the center of our galaxy.”

Previous studies have indeed shown that there are more gamma rays coming from the galactic center than expected, fueling some scientific papers and media reports that suggest the signal might hint at long-sought dark matter particles. However, gamma rays are produced in a number of other cosmic processes, which must be ruled out before any conclusion about dark matter can be drawn. This is particularly challenging because the galactic center is extremely complex, and astrophysicists don’t know all the details of what’s going on in that region.

Most of the Milky Way’s gamma rays originate in gas between the stars that is lit up by cosmic rays, charged particles produced in powerful star explosions called supernovae. This creates a diffuse gamma-ray glow that extends throughout the galaxy. Gamma rays are also produced by supernova remnants, pulsars—collapsed stars that emit “beams” of gamma rays like cosmic lighthouses—and more exotic objects that appear as points of light.

“Two recent studies by teams in the US and the Netherlands have shown that the gamma-ray excess at the galactic center is speckled, not smooth as we would expect for a dark matter signal,” says KIPAC’s Eric Charles, who contributed to the new analysis. “Those results suggest the speckles may be due to point sources that we can’t see as individual sources with the LAT because the density of gamma-ray sources is very high and the diffuse glow is brightest at the galactic center.”

### Remains of ancient stars

The new study takes the earlier analyses to the next level, demonstrating that the speckled gamma-ray signal is consistent with pulsars.

“Considering that about 70 percent of all point sources in the Milky Way are pulsars, they were the most likely candidates,” Di Mauro says. “But we used one of their physical properties to come to our conclusion. Pulsars have very distinct spectra—that is, their emissions vary in a specific way with the energy of the gamma rays they emit. Using the shape of these spectra, we were able to model the glow of the galactic center correctly with a population of about 1,000 pulsars and without introducing processes that involve dark matter particles.”

The team is now planning follow-up studies with radio telescopes to determine whether the identified sources are emitting their light as a series of brief light pulses—the trademark that gives pulsars their name.

Discoveries in the halo of stars around the center of the galaxy, the oldest part of the Milky Way, also reveal details about the evolution of our galactic home, just as ancient remains teach archaeologists about human history.

“Isolated pulsars have a typical lifetime of 10 million years, which is much shorter than the age of the oldest stars near the galactic center,” Charles says. “The fact that we can still see gamma rays from the identified pulsar population today suggests that the pulsars are in binary systems with companion stars, from which they leach energy. This extends the life of the pulsars tremendously.”

### Dark matter remains elusive

The new results add to other data that are challenging the interpretation of the gamma-ray excess as a dark matter signal.

“If the signal were due to dark matter, we would expect to see it also at the centers of other galaxies,” Digel says. “The signal should be particularly clear in dwarf galaxies orbiting the Milky Way. These galaxies have very few stars, typically don’t have pulsars and are held together because they have a lot of dark matter. However, we don’t see any significant gamma-ray emissions from them.”

The researchers believe that a recently discovered strong gamma-ray glow at the center of the Andromeda galaxy, the major galaxy closest to the Milky Way, may also be caused by pulsars rather than dark matter.

But the last word may not have been spoken. Although the Fermi-LAT team studied a large area of 40 degrees by 40 degrees around the Milky Way’s galactic center (the diameter of the full moon is about half a degree), the extremely high density of sources in the innermost four degrees makes it very difficult to see individual ones and rule out a smooth, dark matter-like gamma-ray distribution, leaving limited room for dark matter signals to hide.

This work was funded by NASA and the DOE Office of Science, as well as agencies and institutes in France, Italy, Japan and Sweden.

*Editor's note: A version of this article was originally published by SLAC National Accelerator Laboratory.*

### John Baez - Azimuth

I have a new favorite molecule: adamantane. As you probably know, someone is said to be ‘adamant’ if they are unshakeable, immovable, inflexible, unwavering, uncompromising, resolute, resolved, determined, firm, rigid, or steadfast. But ‘adamant’ is also a legendary mineral, and the etymology is the same as that for ‘diamond’.

The molecule adamantane, shown above, features 10 carbon atoms arranged just like a small portion of a diamond crystal! It’s a bit easier to see this if you ignore the 16 hydrogen atoms and focus on the carbon atoms and bonds between those:

It’s a somewhat strange shape.

**Puzzle 1.** Give a clear, elegant description of this shape.

**Puzzle 2.** What is its symmetry group? This is really two questions: I’m asking about the symmetry group of this shape as an abstract graph, but also the symmetry group of this graph as embedded in 3d Euclidean space, counting both rotations and reflections.

**Puzzle 3.** How many ‘kinds’ of carbon atoms does adamantane have? In other words, when we let the symmetry group of this graph act on the set of vertices, how many orbits are there? (Again this is really two questions, depending on which symmetry group we use.)

**Puzzle 4.** How many kinds of bonds between carbon atoms does adamantane have? In other words, when we let the symmetry group of this graph act on the set of edges, how many orbits are there? (Again, this is really two questions.)

You can see the relation between adamantane and a diamond if you look carefully at a diamond crystal, as shown in this image by H. K. D. H. Bhadeshia:

or this one by Greg Egan:

Even with these pictures at hand, I find it a bit tough to see the adamantane pattern lurking in the diamond! Look again:

Adamantane has an interesting history. The possibility of its existence was first suggested by a chemist named Decker at a conference in 1924. Decker called this molecule ‘decaterpene’, and registered surprise that nobody had made it yet. After some failed attempts, it was first synthesized by the Croatian-Swiss chemist Vladimir Prelog in 1941. He later won the Nobel prize for his work on stereochemistry.

However, long before it was synthesized, adamantane was isolated from petroleum by the Czech chemists Landa, Machacek and Mzourek! They did it in 1932. They only managed to make a few milligrams of the stuff, but we now know that petroleum naturally contains between .0001% and 0.03% adamantane!

Adamantane can be crystallized:

but ironically, the crystals are rather soft. It’s all that hydrogen. It’s also amusing that adamantane has an *odor*: supposedly it smells like camphor!

Adamantane is just the simplest of the molecules called diamondoids.

These are a few:

1 is **adamantane**.

2 is called **diamantane**.

3 is called **triamantane**.

4 is called **isotetramantane**, and it comes in two mirror-image forms.

Here are some better pictures of diamantane:

People have done lots of chemical reactions with diamondoids. Here are some things they’ve done with the next one, **pentamantane**:

Many different diamondoids occur naturally in petroleum. Though the carbon in diamonds is not biological in origin, the carbon in diamondoids found in petroleum is. This was shown by studying ratios of carbon isotopes.

Eric Drexler has proposed using diamondoids for nanotechnology, but he’s talking about larger molecules than those shown here.

For more fun along these lines, try:

• Diamonds and triamonds, *Azimuth*, 11 April 2016.