This post is part of a series on category theory, see Overview of Blog Posts for all posts. All categories are assumed to be locally small, unless stated otherwise.
Previously, we have studied particular examples of limits and colimits. This time, we will go into the general theory. Using the close relation between limits and representable functors and the previous results on universal elements and representability, we are able to provide different perspectives on limits.
Limits as Universal Cones
Definition C7.1 Let be a small category, let be any category, then the diagonal functor is defined as follows:
- For any object , is the constant functor with values in , i.e. the functor which sends every object of to and every morphism in to
- For any morphism in , we define to be the natural transformation that has for any object , the component
Definition C7.2 Let be a small category, let be any category and let be a functor. Then a cone over consists of an object -called apex- and a natural transformation . The naturality condition can be stated by saying that the following diagram commutes for any morphism in :
Given two cones and , a morphism of cones is a morphism such that . Morphisms of cones can be composed and so we obtain a category of cones over , denoted by .
Definition C7.3 A limit is a terminal object in the category of cones .
If the limit is given by a cone consisting of an object and a natural transformation , then we call the limit object (or just limit, by abuse of language) and the components of the structure morphisms.
Remark Being a terminal object, a limit is unique up to unique isomorphism of cones.
Example C7.4 Let be a discrete category, then a functor is nothing but a family of objects . In this case, a limit is just a product of the objects . As a special case, a limit over the empty functor is a terminal object.
Example C7.5 Let be a set and consider the following category , having two objects and for every , a morphism (and of course the identities for and ). There’s a unique way to define composition to turn this into a category. Then having a functor is equivalent to having a family of morphisms in with common source and target. In this case, a limit over is equivalently an equalizer for the family of morphisms .
Example C7.6 Let be the category with three objects and as morphisms (aside from the identities) a unique morphism and a unique morphism . Then a functor is equivalently a triple of objects and two morphisms and . As one might guess, a limit over such a functor is equivalently a pullback .
Example C7.7 Consider with the usual ordering as a category (this is in particular a preorder). Suppose that we have a descending sequence of sets . Then we can define a contravariant functor given by on objects and for two natural numbers with , i.e. a morphism in we let to be subset inclusion . Then the limit of the functor is the intersection , with structure morphisms being the inclusions . Indeed, for a collection of maps from an object , such that for , we have that and agree, we see that all the are equal. Hence the image is contained in and there’s a unique map that makes the necessary triangles commute. Thus we have found a terminal object in the category of cones over .
Definition C7.8 Given a functor from a small category to a category , then can also be considered as a functor , then a cocone over is defined as a cone over . A colimit is defined as an initial object in the category of cocones. Explicitly, a cocone is an object with a natural transformation . The colimit is denoted by .
Remark A limit in is a colimit in and vice versa.
Limits as Adjoints to the Diagonal Functor
Let be a small category, let be any category, then consider the following functor , denoted by .
Suppose that this functor is representable, then take a representation . Then is a limit object of and are the structure morphisms, i.e. is a terminal object in the category of cones. This is just a special case of general theorems we have proved on representable functors in virtue of the following observation:
Remark The category of cones is the opposite category of the category of elements for the functor , i.e.
Proof An object in is an object together with an element , i.e. a cone. A morphism consists of a morphism in the category of elements , i.e. a morphism in , i.e. a morphism in such that , which means that . This is precisely the notion of a morphism of cones.
Now we can apply lemma C5.15 and lemma C5.3, that is, the fact that initial objects in the category of elements for a functor and representations of that functor both correspond to universal elements to conclude the following:
Lemma C7.10 is representable if and only if the limit of in exists and representations of that functor correspond to limit objects together with structure morphisms satisfying the universal property with respect to .
General Limits from Products and Equalizers
General limits can be quite complicated. But apart from some specific examples, we don’t know yet what limits look like in familiar categories such as or if they even exist. We have already seen how pullbacks can be constructed from products and equalizers. (Cf. lemma C6.23) The following construction describes a vast generalization:
Theorem C7.11 Let be a functor from a small category to a category that has all products and binary equalizers. Then exists and can be constructed as follows: Take the product with structure maps . Now for every morphism in , we have a morphism , so that by the universal property of the product, there is a unique morphism such that for all morphisms in . Then the equalizer is the limit together with structure morphisms given by , where is the structure morphism of the equalizer.
Proof We first check that the collection of makes into a cone. Let be a morphism in . Then we have
, showing that the form the components of a natural transformation , that is we have a cone over . Let be a cone over , then for all , we have a morphism . So by the universal property of the product, we get a unique morphism such that for all . Consider the morphism . We have for any morphism in
(where we used naturality of ). So that by uniqueness of , we get that . Thus by the universal property of the equalizer , we get that a unique morphism such that .
To see that is a morphism of cones, note that . To see that is unique, let be another morphism of cones, then satisfies
Furthermore, we have , so that by uniqueness of , we get , i.e. . By the uniqueness part of the universal property of the equalizer, we get that .
This theorem not only gives us a criterion for the existence of all limits in a category from just some specific ones, but it also tells us what they look like, given that we understand products and equalizers:
Example C7.12 Consider the category of sets , then limits can be constructed as follows: be a functor from a small category , then the limit of is given by
Exercise C7.13 Using previously estabished contructions of coproducts and coequalizers in , describe general colimits in .
Limits as Representations of the Limit Functor
We have already seen that for a functor from a small category to any category , the existence of a limit is equivalent to the representability of the functor . In this section we shall derive another description of this functor that relates limits in arbitrary categories to limits in .
Definition C7.14 Let be a small category, be any category and be a functor. Then the functor is defined as follows: for an object , we can consider the functor , (i.e. we compose with the covariant Hom-functor corresponding to ). Since limits in exist, we can take the limit . This is the value of the functor at . (Exercise: Check that this is indeed a functor, i.e. that it acts in a compatible way on morphisms)
Remark This is actually the limit in the functor category. One can check that limits in functor categories may be computed “pointwise”.
Lemma C7.15 There’s a natural isomorphism of functors : Both functors send an object to the set of cones with as an apex.
Proof By definition, an element of is a cone with is an apex. For , we can use the description of example C7.12:
Clearly this set consists of all natural transforations . Naturality is easily checked: Indeed, using the now established identifications on objects with the functor sending an object to the set of cones with as an apex, both functors send a morphism to the map from the set of cones over with as an apex to the set of cones over with as an apex that takes a cone to the cone with as an apex with the natural transformation whose components are for each .
Corollary C7.16 A limit of a functor from a small category to exists if and only if the limit functor is representable.
Corollary C7.17 Covariant Hom-functors are “continuous” in the following sense: if a limit over a functor exists, where is a small category and is any category, then
Corollary C7.18 Dualizing the previous corollary, contravariant Hom-functor send colimits to limits.