nLab
Yoneda lemma for (infinity,1)-categories

Idea

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

Yoneda embedding

Definition

For $C$ an (∞,1)-category and $\mathrm{PSh}(C)$ its (∞,1)-category of (∞,1)-presheaves, the $(\mathrm{\infty },1)$-Yoneda embedding is the (∞,1)-functor

$$y:C\to \mathrm{PSh}(C)$$y : C \to PSh(C)

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

Properties

Yoneda lemma

Theorem

$(\mathrm{\infty },1)$-Yoneda embedding

Let $C$ be an (∞,1)-category and $\mathrm{PSh}(C):=\mathrm{Func}(C^{op},\mathrm{\infty }\mathrm{Grpd})$ be the corresponding (∞,1)-category of (∞,1)-presheaves. Then the canonical (∞,1)-functor

$$Y:C\to \mathrm{PSh}(C)$$Y : C \to PSh(C)

is a full and faithful (∞,1)-functor.

Proof

In terms of quasi-categories, this is proposition 5.1.3.1 in

• Jacob Lurie, Higher Topos Theory

Theorem

$(\mathrm{\infty },1)$-Yoneda theorem

For $C$ a small $(\mathrm{\infty },1)$-category and $F:C^{\mathrm{op}}\to \mathrm{\infty }\mathrm{Grpd}$ an $(\mathrm{\infty },1)$-functor, the composite

$$C^{\mathrm{op}}\to {\mathrm{PSh}}_{(\mathrm{\infty },1)}(C{)}^{\mathrm{op}}\stackrel{\mathrm{Hom}(-,F)}{\to }\mathrm{\infty }\mathrm{Grpd}$$C^{op} \to PSh_{(\infty,1)}(C)^{op} \stackrel{Hom(-,F)}{\to} \infty Grpd

is equivalent to $F$.

Proof

This appears as HTT Lemma 5.5.2.1.

The statement is a direct consequence of the sSet-enriched Yoneda lemma by using the fact that the (∞,1)-category of (∞,1)-presheaves ${\mathrm{PSh}}_{(\mathrm{\infty },1)}(C)$ is modeled by the enriched functor category $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ with $C$ regarded as a simplicially enriched category and using the global model structure on simplicial presheaves.

Preservation of limits

Theorem

The $(\mathrm{\infty },1)$-Yoneda embedding $y:C\to \mathrm{PSh}(C)$ preserves all (∞,1)-limits that exist in $C$.

This appears as HTT, prop. 5.1.3.2.

Local Yoneda embedding

Proposition

For $C$ an (∞,1)-site and $𝒳$ an (∞,1)-topos, (∞,1)-geometric morphisms $(f^{*}⊣f_{*})\mathrm{Sh}(C)\stackrel{\stackrel{f^{*}}{\leftarrow }}{\underset{f_{*}}{\to }}𝒳$ from the (∞,1)-sheaf (∞,1)-topos $\mathrm{Sh}(C)$ to $𝒳$ correspond to the local (∞,1)-functors $f^{*}:C\to 𝒳$, those that

• are left exact (∞,1)-functors;

• send covering families $\{U_i\to X\}$ in $𝒢$ to effective epimorphism

  $$\coprod _i{f}^{*}(U_i)\to f^{*}(X).$$\coprod_i f^*(U_i) \to f^*(X) \,.

More preseicely, the (∞,1)-functor

$$\mathrm{Topos}(𝒳,{\mathrm{Sh}}_{(\mathrm{\infty },1)}(𝒢))\stackrel{L}{\to }\mathrm{Topos}(𝒳,{\mathrm{PSh}}_{(\mathrm{\infty },1)}(𝒢))\stackrel{y}{\to }\mathrm{Func}(𝒢,𝒳)$$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. 6.2.3.20).

References

Published statements appear in

• Jacob Lurie, Higher Topos Theory

as indicated above.

See also the discussion on MathOverflow.