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.



PShPSh can be extended to a functor PSh:(,1)Cat(,1)Cat^PSh : (\infty,1)Cat \to (\infty,1)\widehat{Cat} so that the yoneda embedding CPSh(C)C \to PSh(C) is a natural transformation.

Here, (,1)Cat^(\infty,1)\widehat{Cat} is the (∞,1)-category of large (∞,1)-categories.

This follows from (HTT, prop., together with the identification of PSh(C)PSh(C) with the category obtained by freely adjoining small colimits to CC. This functor is locally left adjoint to the contravariant functor CFunc(C op,Grpd)C \mapsto Func(C^\op, \infty Grpd).

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.


Discussion in the context of an ∞-cosmos:

Last revised on May 24, 2021 at 14:57:33. See the history of this page for a list of all contributions to it.