Yoneda lemma for (infinity,1)-categories


Yoneda lemma

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



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

Yoneda embedding


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).


Yoneda lemma


(,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.


In terms of quasi-categories, this is proposition in


(,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.


This appears as HTT Lemma

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


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.

Local Yoneda embedding


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.


Published statements appear in

as indicated above.

See also the discussion on MathOverflow.

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