nLab Yoneda lemma for higher categories

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 lemma for (∞,1)-categories

• Yoneda lemma for bicategories

• Yoneda lemma for tricategories

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 $C$.

Related concepts

• higher topos theory