Yoneda lemma for (infinity,1)-categories


Yoneda lemma

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



The statement of the Yoneda lemma generalizes 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) \coloneqq 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.

(HTT, prop.


(,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.

(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.

(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.

(HTT, prop.


See also the discussion on MathOverflow.

Revised on March 11, 2016 11:04:47 by Urs Schreiber (