# nLab Yoneda lemma for (infinity,1)-categories

Yoneda lemma

## In higher category theory

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

## 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 $\mathrm{PSh}\left(C\right)$ its (∞,1)-category of (∞,1)-presheaves, the $\left(\infty ,1\right)$-Yoneda embedding is the (∞,1)-functor

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

given by $y\left(X\right):U↦C\left(U,X\right)$.

## Properties

### Yoneda lemma

###### Theorem

$\left(\infty ,1\right)$-Yoneda embedding

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

$Y:C\to \mathrm{PSh}\left(C\right)$Y : C \to PSh(C)
###### Proof

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

###### Theorem

$\left(\infty ,1\right)$-Yoneda theorem

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

${C}^{\mathrm{op}}\to {\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C{\right)}^{\mathrm{op}}\stackrel{\mathrm{Hom}\left(-,F\right)}{\to }\infty \mathrm{Grpd}$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 ${\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)$ is modeled by the enriched functor category $\left[{C}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj}}$ with $C$ regarded as a simplicially enriched category and using the global model structure on simplicial presheaves.

### Preservation of limits

###### Theorem

The $\left(\infty ,1\right)$-Yoneda embedding $y:C\to \mathrm{PSh}\left(C\right)$ 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 $\left({f}^{*}⊣{f}_{*}\right)\mathrm{Sh}\left(C\right)\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}𝒳$ from the (∞,1)-sheaf (∞,1)-topos $\mathrm{Sh}\left(C\right)$ to $𝒳$ correspond to the local (∞,1)-functors ${f}^{*}:C\to 𝒳$, those that

• are left exact (∞,1)-functors;

• send covering families $\left\{{U}_{i}\to X\right\}$ in $𝒢$ to effective epimorphism

$\coprod _{i}{f}^{*}\left({U}_{i}\right)\to {f}^{*}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$\coprod_i f^*(U_i) \to f^*(X) \,.

More preseicely, the (∞,1)-functor

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

as indicated above.