nLab Yoneda lemma for higher categories

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Yoneda lemma

In higher category theory

Higher category theory

higher category theory

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

$y : C \to PSh(C)$

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

Properties

Yoneda lemma

Proposition

$(\infty,n)$-Yoneda embedding

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

$Y : C \to PSh(C)$

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

Proposition

$(\infty,n)$-Yoneda theorem

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

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

is equivalent to $F$.

Preservation of limits

Proposition

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

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