Let be a (locally small) category. Recall that any such category naturally admits a Yoneda embedding
into its presheaf category (where we use to denote the category of functors ). The Yoneda lemma asserts in particular that is full and faithful, which justifies calling it an embedding.
When is in addition assumed to be small, the Yoneda embedding has the following elegant universal property.
Theorem: The Yoneda embedding exhibits as the free cocompletion of in the sense that for any cocomplete category , the restriction functor
from the category of cocontinuous functors to the category of functors is an equivalence. In particular, any functor extends (uniquely, up to natural isomorphism) to a cocontinuous functor , and all cocontinuous functors arise this way (up to natural isomorphism).
Colimits should be thought of as a general notion of gluing, so the above should be understood as the claim that is the category obtained by “freely gluing together” the objects of in a way dictated by the morphisms. This intuition is important when trying to understand the definition of, among other things, a simplicial set. A simplicial set is by definition a presheaf on a certain category, the simplex category, and the universal property above says that this means simplicial sets are obtained by “freely gluing together” simplices.
In this post we’ll content ourselves with meandering towards a proof of the above result. In a subsequent post we’ll give a sampling of applications.