nLab
Yoneda lemma for (infinity,1)-categories

Contents

Context

Yoneda lemma

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

Contents

Idea

The statement of the Yoneda lemma generalizes 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 \colon C \to PSh(C)

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

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

Properties

Yoneda lemma

Proposition

(,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.

For small \infty-categories this is HTT, prop. 5.1.3.1. For large \infty-categories see Riehl & Verity 2018, Thm. 7.2.22 (which considers \infty-presheaves regarded under the \infty -Grothendieck construction).

Proposition

(,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 5.5.2.1)

Proof

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.

Naturality

Proposition

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. 5.3.6.10), 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

Proposition

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

(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.

(HTT, prop. 6.2.3.20)

References

Discussion in the context of an ∞-cosmos:

Last revised on October 9, 2021 at 14:26:35. See the history of this page for a list of all contributions to it.