nLab 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 \colon C \to PSh(C)

given by y(X):UC(U,X)y(X) \colon U \mapsto C(U,X).

Seen under the \infty -Grothendieck construction this is Riehl & Verity 2018, Def. 6.2.3.


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 \colon C \to PSh(C)

is a full and faithful (∞,1)-functor.

For small \infty-categories this is HTT, prop. For possibly large \infty-categories see Riehl & Verity 2018, Thm. 7.2.22 (which considers \infty-presheaves regarded under the \infty -Grothendieck construction) and Kerodon, Thm.


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

For CC a small (,1)(\infty,1)-category and F:C opGrpdF \colon 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.

For small \infty-sites this is HTT, Lemma For possibly large \infty-sites see Kerodon, Prop.


For small \infty-sites, the statement may be obtained as a 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 \colon (\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(𝒳,𝒢) Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G})) \stackrel{L}{\to} Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G})) \stackrel{y}{\to} Func(\mathcal{X}, \mathcal{G})

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:

Discussion internal to any (∞,1)-topos:

Formalization of the (,1)(\infty,1)-Yoneda lemma via simplicial homotopy type theory (in Rzk):

Last revised on May 15, 2024 at 06:58:32. See the history of this page for a list of all contributions to it.