nLab
Yoneda lemma for (infinity,1)-categories

Context

Yoneda lemma

(,1)(\infty,1)-Category theory

Contents

Idea

The statement of the Yoneda lemma has a straightforward generalization from categories to (∞,1)-categories.

Yoneda embedding

Definition

For CC an (∞,1)-category and PSh(C)PSh(C) its (∞,1)-category of (∞,1)-presheaves, the (,1)(\infty,1)-Yoneda embedding is the (∞,1)-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

Theorem

(,1)(\infty,1)-Yoneda embedding

Let CC be an (∞,1)-category and PSh(C):=Func(C op,Grpd)PSh(C) := Func(C^\op, \infty Grpd) be the corresponding (∞,1)-category of (∞,1)-presheaves. Then the canonical (∞,1)-functor

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

is a full and faithful (∞,1)-functor.

Proof

In terms of quasi-categories, this is proposition 5.1.3.1 in

Theorem

(,1)(\infty,1)-Yoneda theorem

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

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

is equivalent to FF.

Proof

This appears as HTT Lemma 5.5.2.1.

The statement is a direct consequence of the sSet-enriched Yoneda lemma by using the fact that the (∞,1)-category of (∞,1)-presheaves PSh (,1)(C)PSh_{(\infty,1)}(C) is modeled by the enriched functor category [C op,sSet] proj[C^{op}, sSet]_{proj} with CC regarded as a simplicially enriched category and using the global model structure on simplicial presheaves.

Preservation of limits

Theorem

The (,1)(\infty,1)-Yoneda embedding y:CPSh(C)y : C \to PSh(C) preserves all (∞,1)-limits that exist in CC.

This appears as HTT, prop. 5.1.3.2.

Local Yoneda embedding

Proposition

For CC an (∞,1)-site and 𝒳\mathcal{X} an (∞,1)-topos, (∞,1)-geometric morphisms (f *f *)Sh(C)f *f *𝒳(f^* \dashv f_*) Sh(C) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{X} from the (∞,1)-sheaf (∞,1)-topos Sh(C)Sh(C) to 𝒳\mathcal{X} correspond to the local (∞,1)-functors f *:C𝒳f^* : C \to \mathcal{X}, those that

More preseicely, the (∞,1)-functor

Topos(𝒳,Sh (,1)(𝒢))LTopos(𝒳,PSh (,1)(𝒢))yFunc(𝒢,𝒳) Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G})) \stackrel{L}{\to} Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G})) \stackrel{y}{\to} Func(\mathcal{G}, \mathcal{X})

given by precomposition of inverse image functors by ∞-stackification and by the (∞,1)-Yoneda embedding is a full and faithful (∞,1)-functor and its essential image is spanned by these local morphisms.

This appears as (HTT, prop. 6.2.3.20).

References

Published statements appear in

as indicated above.

See also the discussion on MathOverflow.

Revised on January 2, 2011 12:14:59 by Urs Schreiber (82.113.121.0)