# Presheaves are colimits of representables (topologies 2/6)

Last week, I started a series on sieves and Grothendieck topoi. This is a short intermezzo on a well-known lemma from category theory that I will need for next week’s instalment.

Let be a small category. Recall that a presheaf on is a functor . Examples include the representable presheaves for , given by . The Yoneda lemma says that for any presheaf and any , the map

is an isomorphism. Applying this to shows that the Yoneda embedding

is fully faithful.

Given a functor , write for the comma category whose objects are pairs where and is a morphism (natural transformation) in . A morphism is a morphism such that the triangle

(1)

of natural transformations commutes, where denotes postcomposition by . Note that is again a small category, and there is a forgetful functor taking to .

By the Yoneda lemma, the category is isomorphic (not just equivalent!) to the category of pairs with and with morphisms given by morphisms in such that . It’s convenient to keep both points of view.

Lemma. Let be a small category, and let be a functor.

1. The object is naturally a cocone under via the morphisms .
2. This cocone makes the colimit of the diagram of representable functors.

In particular, any presheaf on a small category is a colimit of representable presheaves.

Proof. A cocone under is a presheaf with a natural transformation to the constant diagram with value . This means every is taken to a natural transformation , such that for any morphism , the square

commutes. By the Yoneda lemma, such a datum corresponds to an association of elements for all such that for every with , we have .

(1) To make a cocone under , simply take .

(2) Given any other cocone under , define the natural transformation by

Naturality follows since implies . It is clear that is the unique natural transformation of cocones under , showing that is the colimit.

One can also easily rewrite this argument in terms of natural transformations . For instance, the universal cocone is the natural transformation of functors given on by . Naturality of follows at once from (1). But checking that this thing is universal is a bit more tedious in this language.

Example. A standard example where this point of view is useful is simplicial sets. Let be the category of finite nonempty totally ordered sets, with (weakly) monotone increasing functions as morphisms. A simplicial set is a functor , and we often think of them as combinatorial models for topological spaces. The representable ones are the standard -simplices , where is the totally ordered set for .

If is a simplicial set, its value at is called the -simplices of . By the Yoneda lemma, this is . Then the story above is saying that a simplicial set is the colimit over all its -simplices for . This is extremely useful, as many arguments proceed by attaching simplices one at a time.

# Simplicial sets

A few weeks ago, I finally struck up the courage to take some baby steps reading Lurie’s Higher topos theory. In a series of posts mostly written for my own benefit, I will untangle some of the basic definitions and provide some easy examples. The first one is one I was already somewhat familiar with: simplicial sets.

Definition. For each , write for the poset . The full subcategory of on these is denoted , the simplex category. Concretely, it has objects for all , and morphisms

A simplicial set is a functor . This can be described rather concretely using the objects and the face and degeneracy maps between them; see e.g. Tag 0169. The category of simplicial sets is usually denoted , , or (in analogy with cosimplicial sets ).

The representable simplicial set is usually denoted or . Then the Yoneda lemma shows that the functor given by is represented by , i.e.

Definition. The geometric realisation functor is defined as follows: for , the geometric realisation is the standard -simplex

(If no confusion arises, it may also be denoted .) This is functorial in : for a map (equivalently, by the Yoneda lemma, a map ) we get a continuous map by

For an arbitrary simplicial set , write

where the transition map corresponding to a map over is defined via

This is functorial in , and when it coindices with the previous definition because the identity is terminal in the index category.

Remark. In a fancier language, is the left Kan extension of the functor along the Yoneda embedding . (Those of you familiar with presheaves on spaces will recognise the similarity with the definition of for a continuous map of topological spaces, which is another example of a left Kan extension.)

Remark. It is a formal consequence of the definitions that geometric realisation preserves arbitrary colimits (“colimits commute with colimits”). This also follows because it is a left adjoint to the singular set functor, but we won’t explore this here.

Wisdom. The most geometric way to think about a simplicial set is through its geometric realisation.

For example, we can define the horn in as the union of the images of the maps coming from the face maps for . Since geometric realisation preserves colimits (alternatively, stare at the definitions), we see that the geometric realisation of is obtained in the same way from the maps , so it is the -simplex with its interior and face opposite the vertex removed.

The geometric realisation is a good first approximation for thinking about a simplicial set. However, when thinking about -categories (e.g. in the next few posts), this is actually not the way you want to think about a simplicial set. Indeed, homotopy of simplicial sets (equivalently their geometric realisations) is stronger than equivalence of -categories. (More details later, hopefully.)