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.
(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.
- The object is naturally a cocone under via the morphisms .
- 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.