Guest post by Christina Vasilakopoulou
In the eighth installment of the Kan Extension Seminar, we discuss
the paper “Elementary Observations on 2-Categorical Limits” by G.M. Kelly, published in 1989. Even though Kelly’s classic book Basic Concepts of Enriched Category Theory, which contains the abstract theory related to indexed (or weighted) limits for arbitrary -categories, was available since 1982, the existence of the present article is well-justifiable.
On the one hand, it constitutes an independent account of the fundamental case =, thus it motivates and exemplifies the more general framework through a more gentle, yet meaningful exposition of 2-categorical limits. The explicit construction of specific notable finite limits such as inserters, equifiers etc. promotes the comprehension of the definitions, via a hands-on description. Moreover, these finite limits and particular results concerning 2-categories rather than general enriched categories, such as the construction of the cotensor as a PIE limit, are central for the theory of 2-categories. Lastly, by introducing indexed lax and pseudo limits along with Street’s bilimits, and providing appropriate lax/ pseudo/ bicategorical completeness results, the paper serves also as an indespensable reference for the later “2-Dimensional Monad Theory” by Blackwell, Kelly and Power.
I would like to take this opportunity to thank Emily as well as all the other participants of the Kan Extension Seminar. This has been a unique experience of constant motivation and inspiration for me!
Presently, our base of enrichment is the cartesian monoidal closed category of (small) categories, with the usual adjunction .
The very definition of an indexed limit requires a good command
of the basic -categorical notions, as seen for example in
“Review of the Elements of 2-categories” by Kelly and Street. In particular, a 2-natural transformation between 2-functors consists of components which not only satisfy the usual naturality condition, but also the 2-naturality one expressing compatibility with 2-cells. Moreover, a modification between 2-natural transformations has components families of 2-cells compatible with the mapped 1-cells of the domain 2-category, i.e. (where is whiskering).
A 2-functor is called representable, when there exists a 2-natural isomorphism
The components of this isomorphism are in , and the unit of the representation is the corresponding `element’ via Yoneda.
For a general complete symmetric monoidal closed category , the usual functor category for two -categories is endowed with the structure of a -category itself, with hom-objects ends
(which exist at least when is small). In our context of = it is not necessary to employ ends and coends at all, and the hom-category of the functor 2-category is evidently the category of 2-natural transformations and modifications. However, we note that computations via (co)ends simplify and are essential for constructions and (co)completeness results for enrichment in general monoidal categories.
The definition of weighted limits for 2-categories
To briefly motivate the definition of a weighted limit, recall that an ordinary limit of a (-) functor is characterized by an isomorphism
natural in , where is the constant functor on the object . In other words, the limit is the representing object of the presheaf
Since components of a natural transformation (i.e. cones) can be viewed as components of a natural
, the above defining isomorphism can be written as
In this form, ordinary limits can be easily seen as particular examples of conical indexed limits for =, and we are able to generalize the concept of a limit by replacing the functor by an arbitrary functor (weight) .
We may thus think of a 2-functor as a (small) indexing type or weight, and a 2-functor as a diagram in of shape :
The 2-functor gives rise to a 2-functor
which maps a 0-cell to the category . A representation of this contravariant 2-functor is an object along with 2-natural isomorphism
with components isomorphisms between categories
The unit of this representation is
which corresponds uniquely to a 2-natural transformation
Via this 2-natural isomorphism, the object in satisfies a universal property which can be expressed in two levels:
The 1-dimensional aspect of the universal property states that every natural transformation factorizes as
for a unique 1-cell in , where the vertical arrow is just pre-composition with .
The 2-dimensional aspect of the universal property states that every modification factorizes as for a unique 2-cell in .
The fact that the 2-dimensional aspect (which asserts an isomorphism of categories) does not in general follow from the 1-dimensional aspect (which asserts a bijection between the hom-sets of the underlying categories) is a recurrent issue of the paper. In fact, things would be different if the underlying category functor
were conservative, in which case the 2-dimensional universal property would always imply the 1-dimensional one. Certainly though, this is not the case for =: the respective functor discards all the 2-cells and is not even faithful. However, if we know that a weighted limit exists, then the first level of the universal property suffices to detect it up to isomorphism.
Completeness of 2-categories
A 2-category is complete when all limits exist. The defining 2-natural isomorphism extends the mapping into a functor of two variables (the weighted limit functor)
as the left parametrized adjoint (actually its opposite) of the functor
mapping an object and a functor to .
A colimit in is a limit in , and the weighted colimit functor is
Apart from the evident duality, we observe that often colimits are harder to compute than limits. This may partially be due to the fact that is determined by the representable which gives generalized elements of , whereas the description of
gives us arrows out of . For example, limits in are easy to compute via
and in particular, taking gives us the objects of the category and gives us the morphisms. On the contrary, colimits in are not straightforward (except than their property ).
Notice that like ordinary limits are defined, via representability, in terms of limits in , we can define weighted limits in terms of limits of representables in :
On the other hand, if the weights are representables, via Yoneda lemma we get
The main result for general -completeness in Kelly’s book says that a -enriched category is complete if and only if it admits all conical limits (equivalently, products and equalizers) and cotensor products. Explicitly, conical limits are those with weight the constant -functor , whereas cotensors are those where the domain enriched category is the unit category , hence the weight and the diagram are determined by objects in and respectively. Once again, for = an elementary description of both limits is possible.
Notice that when a 2-category admits tensor products of the form , then the 2-dimensional universal property follows from the 1-dimensional for every limit, because of conservativity of the functor and the definition of tensors. Moreover, the former also implies that the category is a strong generator in , hence the existence of only the cotensor along with conical limits in a 2-category is enough to deduce 2-completeness.
itself has cotensor and tensor products, given by
and . It is ultimately also cocomplete, all colimits being constructed from tensors and ordinary colimits in (which give the conical limits in by the existence of the cotensor ).
If we were to make use of ends and coends, the explicit construction of an arbitrary 2-(co)limit in as the (co)equalizer of a pair of arrows between (co)products of (co)tensors coincides with
Such an approach simplifies the proofs of many useful properties of limits and colimits, such as
for appropriate 2-functors.
Famous finite 2-limits
The paper provides the description of some important classes of limits in 2-categories, essentially by exhibiting the unit of the defining representation for each particular case. A table which summarizes the main examples included is the following:
Let’s briefly go through the explicit construction of an inserter in a 2-category . The weight and diagram shape are as in the first line of the above table, and denote by the image of the diagram in . The standard technique is to identify the form of objects and morphisms of the functor 2-category , and then state both aspects of the universal property.
An object is a 2-natural transformation with components and satisfying the usual naturality condition
(2-naturality follows trivially, since only has the identity 2-cell). This amounts to the following data:
an 1-cell , i.e. the object in determined by the functor ;
a 2-cell , i.e. the morphism in determined by the functor ;
properties, which make the 1-cells
factorize as and .
We can encode the above data by a diagram
Now a morphism is a modification between two objects as above. This has components
given by 2-cells
satisfying naturality .
The modification condition and gives the components of as whiskered composites of . We can thus express such a modification as a 2-cell satisfying (graphically expressed by pasting accordingly to the sides of ).
This encoding simplifies the statement of the universal property for , as the object of in through which any natural transformation and modification uniquely factorize in an appropriate way (in fact, through the unit ). A very similar process can be followed for the identification of the other classes of limits. As an illustration, let’s consider some of these limits in the 2-category .
The inserter of two functors is a category with objects pairs where and in . A morphism is an arrow in such that the following diagram commutes:
The functor is just the forgetful functor, and the natural transformation is given by .
The comma-object of two functors is precisely the comma category. If the functors have also the same domain, their inserter is a subcategory of the comma category.
The equifier of two natural transformations is the full subcategory of over all objects such that in .
There is a variety of constructions of new classes of limits from given ones, coming down to the existence of endo-identifiers, inverters, iso-inserters, comma-objects, iso-comma-objects, lax/ oplax/pseudo limits of arrows and the cotensors , out of inserters, equifiers and binary products in the 2-category . Along with the substantial construction of arbitrary cotensors out of these three classes, P(roducts)I(nserters)E(quifiers) limits are established as essential tools, also relatively to categories of algebras for 2-monads. Notice that equalizers are `too tight’ to fit in certain 2-categories of importance such as .
Weaker notions of limits in 2-categories
The concept of a weighted 2-limit strongly depends on the specific structure of the 2-category of 2-functors, 2-natural transformations and modifications, for the 2-categories and . If we alter this structure by considering lax natural transformations or pseudonatural transformations, we obtain the definition of the lax limit and pseudo limit as the representing objects for the 2-functors
Notice that the functor categories and are 2-categories whenever is a 2-category, hence the defining isomorphisms are again between categories as before.
An important remark is that any lax or pseudo limit in can be in fact expressed as a `strict’ weighted 2-limit. This is done by replacing the original weight with its image under the left adjoint of the incusion functors , . The opposite does not hold: for example, inserters and equifiers are neither lax not pseudo limits.
We can relax the notion of limits in 2-categories even further, and define the bilimit of 2-functors and as the representing object up to equivalence:
This is of course a particular case of general bilimits in bicategories, for which and are requested to be bicategories and and homomorphism of bicategories. The above equivalence of categories expresses a birepresentation of the homomorphism .
Evidently, bilimits (firstly introduced by Ross Street) may exist even when pseudo limits do not, since they require an equivalence rather than isomorphism of hom-categories. The following two results sum up the conditions ensuring whether a 2-category has all lax, pseudo and bilimits.
A 2-category with products, inserters and equifiers has all lax and pseudo limits (whereas it may not have all strict 2-limits).
A 2-category with biproducts, biequalizers and bicotensors is bicategorically complete. Equivalently, it admits all bilimits if and only if for all 2-functors , from a small ordinary category , the above mentioned birepresentation exists.
Street’s construction of an arbitrary bilimit requires a descent object of a 3-truncated bicosimplicial object in . An appropriate modification of the arguments exhibits lax and pseudo limits as PIE limits.
These weaker forms of limits in 2-categories are fundamental for the theory of 2-categories and bicategories. Many important constructions such as the Eilenberg-Moore object as well as the Grothendieck construction on a fibration, arise as lax/oplax limits. They are also crucial in 2-monad theory, for example when studying categories of (strict) algebras with non-strict (pseudo or even lax/oplax) morphisms, which are more common in nature.