nLab
Yoneda lemma for higher categories

Contents

Context

Yoneda lemma

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

One expects the Yoneda lemma to generalize to essentially every flavor of higher category theory. Various special cases have been (defined and) proven, such as the:

Yoneda embedding

Definition

For CC an (∞,n)-category and PSh(C)Func(C op,(,(n1))Cat)PSh(C)\coloneqq Func(C^\op, (\infty,(n-1)) Cat) its (∞,n)-category of (∞,n)-presheaves?, the (,n)(\infty,n)-Yoneda embedding is the (∞,n)-functor

y:CPSh(C) y : C \to PSh(C)

given by y(X):UC(U,X)y(X) : U \mapsto C(U,X).

Properties

Yoneda lemma

Proposition

(,n)(\infty,n)-Yoneda embedding

Let CC be an (∞,n)-category and PSh(C)PSh(C) be the corresponding (∞,n)-category of (∞,n)-presheaves?. Then the canonical (∞,n)-functor

Y:CPSh(C) Y : C \to PSh(C)

is a full and faithful (∞,n)-functor?.

Proposition

(,n)(\infty,n)-Yoneda theorem

For CC a small (,n)(\infty,n)-category and F:C op(,(n1))CatF : C^{op} \to (\infty,(n-1)) Cat an (,n)(\infty,n)-functor, the composite

C opPSh (,1)(C) opHom(,F)(,(n1))Cat C^{op} \to PSh_{(\infty,1)}(C)^{op} \stackrel{Hom(-,F)}{\to} (\infty,(n-1)) Cat

is equivalent to FF.

Preservation of limits

Proposition

The (,n)(\infty,n)-Yoneda embedding y:CPSh(C)y : C \to PSh(C) preserves all (∞,n)-limit?s that exist in CC.

Last revised on April 15, 2021 at 13:23:22. See the history of this page for a list of all contributions to it.