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

Contents

### Context

#### Yoneda lemma

Yoneda lemma

Ingredients

Incarnations

Properties

Universal aspects

Classification

Induced theorems

In higher category theory

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# Contents

## Idea

The statement of the Yoneda lemma generalizes from categories to (∞,1)-categories.

## Yoneda embedding

###### Definition

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

$y \colon C \to PSh(C)$

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

## Properties

### Yoneda lemma

###### Proposition

$(\infty,1)$-Yoneda embedding

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

$Y \colon C \to PSh(C)$

For small $\infty$-categories this is HTT, prop. 5.1.3.1. 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. 8.2.5.4.

###### Proposition

$(\infty,1)$-Yoneda theorem

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

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

is equivalent to $F$.

For small $\infty$-sites this is HTT, Lemma 5.5.2.1. For possibly large $\infty$-sites see Kerodon, Prop. 8.2.1.3.

###### Proof

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 $PSh_{(\infty,1)}(C)$ is modeled by the enriched functor category $[C^{op}, sSet]_{proj}$ with $C$ regarded as a simplicially enriched category and using the global model structure on simplicial presheaves.

### Naturality

###### Proposition

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

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

This follows from (HTT, prop. 5.3.6.10), together with the identification of $PSh(C)$ with the category obtained by freely adjoining small colimits to $C$. This functor is locally left adjoint to the contravariant functor $C \mapsto Func(C^\op, \infty Grpd)$.

### Preservation of limits

###### Proposition

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

### Local Yoneda embedding

###### Proposition

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

• are left exact (∞,1)-functors;

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

$\coprod_i f^*(U_i) \to f^*(X) \,.$

More preseicely, the (∞,1)-functor

$Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G})) \stackrel{L}{\to} Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G})) \stackrel{y}{\to} Func(\mathcal{X}, \mathcal{G})$

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.

## References

Discussion in the context of an ∞-cosmos:

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

Formalization of the $(\infty,1)$-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.