∞-Yoneda Embedding
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In mathematics, especially
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, the 2-Yoneda lemma is a generalization of the
Yoneda lemma In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a ...
to 2-categories. Precisely, given a contravariant pseudofunctor F on a category ''C'', it says: for each object x in ''C'', the natural functor (evaluation at the identity) :\underline(h_x, F) \to F(x) is an
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of cate ...
, where \underline(-, -) denotes (roughly) the category of natural transformations between pseudofunctors on ''C'' and h_x = \operatorname(-, x). Under the Grothendieck construction, h_x corresponds to the
comma category In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a Category (mathematics), category to one another ...
C \downarrow x. So, the lemma is also frequently stated as: :F(x) \simeq \underline(C \downarrow x, F), where F is identified with the fibered category associated to F. As an application of this lemma, the coherence theorem for bicategories holds.


Sketch of proof

First we define the functor in the opposite direction :\mu : F(x) \to \underline(h_x, F) as follows. Given an object \overline in F(x), define the natural transformation :\mu(\overline) : h_x \to F, that is, \mu(\overline)_y : \operatorname(y, x) \to F(y), by :\mu(\overline)_y(f) = (Ff) \overline. (In the below, we shall often drop a subscript for a natural transformation.) Next, given a morphism \varphi : \overline \to x' in F(x), for f : y \to x, we let \mu(\varphi)(f) be :(Ff)\varphi : (Ff)\overline \to (Ff)x'. Then \mu(\varphi) :\mu(\overline) \to \mu(x') is a morphism (a 2-morphism to be precise or a
modification Modification may refer to: * Modifications of school work for students with special educational needs * Modifications (genetics), changes in appearance arising from changes in the environment * Posttranslational modifications, changes to prote ...
in the terminology of Bénabou). The rest of the proof is then to show # The above \mu is a functor, # e \circ \mu \simeq \operatorname, where e is the evaluation at the identity; i.e., e(\lambda) = \lambda(\operatorname_x), e(\alpha : \lambda \to \rho) = \alpha(\operatorname_x) : \lambda(\operatorname_x) \to \rho(\operatorname_x), # \mu \circ e \simeq \operatorname. Claim 1 is clear. As for Claim 2, :e(\mu(\overline)) = \mu(\overline)(\operatorname_x) = F(\operatorname_x) \overline \simeq \operatorname_ \overline = \overline where the isomorphism here comes from the fact that F is a pseudofunctor. Similarly,e(\mu(\varphi)) \simeq \varphi. For Claim 3, we have: :\mu(e(\lambda))(f) = (Ff \circ \lambda)(\operatorname_x) \simeq (\lambda \circ h_x f)(\operatorname_x) = \lambda(f). Similarly for a morphism \alpha : \lambda \to \rho. \square


∞-Yoneda

Given an
∞-category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a Category (ma ...
''C'', let \widehat = \underline(C^, \textbf) be the ∞-category of presheaves on it with values in Kan = the
∞-category of Kan complexes In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...
. Then the ∞-version of the Yoneda embedding C \hookrightarrow \widehat involves some (harmless) choice in the following way. First, we have the
hom-functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...
:\operatorname : C^ \times C \to \textbf 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 \operatorname\underline(C \times C^, \textbf). Fix one such functor. Then we get the Yoneda embedding functor in the usual way: :y : C \to \widehat, \, a \mapsto \operatorname(-, a), which turns out to be fully faithful (i.e., an equivalence on the Hom level). Moreover and more strongly, for each object F in \widehat and object a in C, the evaluation e at the identity (see below) :\operatorname(y(a), F) \to F(a) is invertible in the ∞-category of large Kan complexes (i.e., Kan complexes living in a universe larger than the given one). Here, the evaluation map e refers to the composition :\operatorname(y(a), F) \overset\to \operatorname(y(a)(a), F(a)) = \operatorname(\operatorname(a, a), F(a)) \to F(a) where the last map is the restriction to the identity \operatorname_a. The ∞-Yoneda lemma is closely related to the matter of straightening and unstraightening.


Notes


References

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Further reading

* * * * * * E. Riehl and D. Verity, Fibrations and Yoneda’s lemma in an ∞-cosmos, J. Pure Appl. Algebra 221 (2017), no. 3, 499–564, arXiv:1506.05500 * https://math.stackexchange.com/questions/1293920/yoneda-lemma-for-2-categories-lax-version * * * *{{Cite web , title=2.2 The Theory of 2-Categories , url=https://kerodon.net/tag/007K , website=Kerodon, ref={{harvid, 8.3.3 Hom-Functors for ∞-Categories in Kerodon Category theory