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2-Yoneda lemma

Result in category theory From Wikipedia, the free encyclopedia

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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:[1] 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:[2]

where is identified with the fibered category associated to .

As an application of this lemma, the coherence theorem for bicategories holds.

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Sketch of proof

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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

  1. The above is a functor,
  2. , where is the evaluation at the identity; i.e.,

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

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∞-Yoneda

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Given an ∞-category C, let be the ∞-category of presheaves on it with values in Kan = the ∞-category of Kan complexes. Then the ∞-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 [3][4] 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).[5] Moreover and more strongly, for each object in and object in , the evaluation at the identity (see below)

is invertible in the ∞-category of large Kan complexes (i.e., Kan complexes living in a universe larger than the given one).[6] Here, the evaluation map refers to the composition

where the last map is the restriction to the identity .[7]

The ∞-Yoneda lemma is closely related to the matter of straightening and unstraightening.

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Notes

References

Further reading

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