In mathematics, especially category theory, the 2-Yoneda lemma is a generalization of the Yoneda lemma to 2-categories. Precisely, given a contravariant pseudofunctor on a category C, it says: for each object in C, the natural functor (evaluation at the identity)
is an equivalence of categories, where denotes (roughly) the category of natural transformations between pseudofunctors on C and .
Under the Grothendieck construction, corresponds to the comma category . So, the lemma is also frequently stated as:
where is identified with the fibered category associated to .
As an application of this lemma, the coherence theorem for bicategories holds.
First we define the functor in the opposite direction
as follows. Given an object in , define the natural transformation
that is, by
(In the below, we shall often drop a subscript for a natural transformation.) Next, given a morphism in , for , we let be
Then is a morphism (a 2-morphism to be precise or a modification in the terminology of Bénabou). The rest of the proof is then to show
Claim 1 is clear. As for Claim 2,
where the isomorphism here comes from the fact that is a pseudofunctor. Similarly, For Claim 3, we have:
Similarly for a morphism
Given an âÂÂ-category C, let be the âÂÂ-category of presheaves on it with values in Kan = the âÂÂ-category of Kan complexes. Then the âÂÂ-version of the Yoneda embedding involves some (harmless) choice in the following way.
First, we have the hom-functor
that is characterized by a certain universal property (e.g., universal left fibration) and is unique up to a unique isomorphism in the homotopy category Fix one such functor. Then we get the Yoneda embedding functor in the usual way:
which turns out to be fully faithful (i.e., an equivalence on the Hom level). Moreover and more strongly, for each object in and object in , the evaluation at the identity (see below)
is invertible in the âÂÂ-category of large Kan complexes (i.e., Kan complexes living in a universe larger than the given one). Here, the evaluation map refers to the composition
where the last map is the restriction to the identity .
The âÂÂ-Yoneda lemma is closely related to the matter of straightening and unstraightening.