# nLab (infinity,1)-category of (infinity,1)-presheaves

### Context

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

(∞,1)-category theory

## Models

#### $(\infty,1)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Definition

For ∞Grpd the (∞,1)-category of ∞-groupoids, and for $S$ a (∞,1)-category (or in fact any simplicial set), an $(\infty,1)$-presheaf on $S$ is an $(\infty,1)$-functor

$F : S^{op} \to \infty Grpd \,.$

The $(\infty,1)$-category of $(\infty,1)$-presheaves is the (∞,1)-category of (∞,1)-functors

$PSh_{(\infty,1)}(S) := Func(S^{op}, \infinity Grpd) \,.$

## Properties

### Models

A model for an $(\infty,1)$-presheaf categories is the model structure on simplicial presheaves. See also the discussion at models for ∞-stack (∞,1)-toposes.

###### Proposition

For $C$ a simplicially enriched category with Kan complexes as hom-objects, write $[C^{op}, sSet_{Quillen}]_{proj}$ and $[C^{op}, sSet_{Quillen}]_{inj}$ for the projective or injective, respectively, gloabl model structure on simplicial presheaves. Write $(-)^\circ$ for the full sSet-enriched subcategory on fibrant-cofibrant objects, and $N(-)$ for the homotopy coherent nerve that sends a Kan-complex enriched category to a quasi-category.

Then there is an equivalence of quasi-categories

$PSh(N(C)) \simeq N ([C^{op}, sSet_{Quillen}]_{proj})^\circ \,.$

Similarly for the injective model structure.

###### Proof

This is a special case of the more general statement that the model structure on functors models an (∞,1)-category of (∞,1)-functors. See there for more details.

Notice that the result in particular means that any $(\infty,1)$-presheaf – an “$\infty$-pseudofunctor” – may be straightened or rectified to a genuine sSet-enriched functor, that respects horizontal compositions strictly.

### Limits and colimits

In an ordinary category of presheaves, limits and colimits are computed objectwise, as described at limits and colimits by example. The analogous statement is true for (∞,1)-limits and colimits in an $(\infty,1)$-category of $(\infty,1)$-presheaves.

This is a special case of the general existence of limits and colimits in an (∞,1)-category of (∞,1)-functors. See there for more details.

###### Corollary

For $C$ a small $(\infty,1)$-category, the $(\infty,1)$-category $PSh(C)$ admits all small limits and colimits.

See around HTT, cor. 5.1.2.4.

### As the free completion under colimits

An ordinary category of presheaves on a small category $C$ is the free cocompletion of $C$, the free completion under forming colimits.

The analogous result holds for $(\infty,1)$-category of $(\infty,1)$-presheaves.

###### Lemma

Let $C$ be a small quasi-category and $j : S \to PSh(C)$ the (∞,1)-Yoneda embedding.

The identity (∞,1)-functor $Id : PSh(C) \to PSh(C)$ is the left (∞,1)-Kan extension of $j$ along itself.

###### Proof

This is HTT, lemma 5.1.5.3.

For $D$ a quasi-category with all small colimits, write $Func^L(PSh(C),D) \subset Func(PSh(C),D)$ for the full sub-quasi-category of the (∞,1)-category of (∞,1)-functors on those that preserve small colimits.

###### Lemma

Composition with the Yoneda embedding $j : C \to PSh(C)$ induces an equivalence of quasi-categories

$Func^L(PSh(C),D) \to Func(C,D) \,.$
###### Proof

This is HTT, theorem 5.1.5.6.

In terms of the model given by the model structure on simplicial presheaves, this is statement made in

which gives that article its name.

###### Definition

Let $A$ and $B$ be model categories, $D$ a plain category and

$\array{ D &\stackrel{r}{\to}& A \\ & \searrow_\gamma \\ && B }$

two plain functors. Say that a model-category theoretic factorization of $\gamma$ through $A$ is

1. a Quillen adjunction $(L \dashv R) : A \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} B$

2. a natural weak equivalence $\eta : L \circ r \to \gamma$

$\array{ D &&\stackrel{r}{\to}&& A \\ & \searrow_\gamma &{}^\eta\swArrow& \swarrow_L \\ && B } \,.$

Let the category of such factorizations have morphisms $((L \dashv R), \eta ) \to ((L' \dashv R'), \eta' )$ given by natural transformations $L \to L'$ such that for all all objects $d \in D$ the diagrams

$\array{ L\circ r(d) &&\to&& L'\circ r(d) \\ & {}_{\eta_{d}}\searrow && \swarrow_{\eta'_{d}} \\ && \gamma() }$

commutes.

Notice that the (∞,1)-category presented by a model category – at least by a combinatorial model category – has all (∞,1)-categorical colimits, and that the Quillen left adjoint functor $L$ presents, via its derived functor, a left adjoint (∞,1)-functor that preserves $(\infty,1)$-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving $(\infty,1)$-functors into $(\infty,1)$-categories that have all colimits.

###### Theorem

(model category presentation of free $(\infty,1)$-cocompletion)

For $C$ a small category, the projective global model structure on simplicial presheaves $[C^{op}, sSet]_{proj}$ on $C$ is universal with respect to such factorizations of functors out of $C$:

every functor $C \to B$ to any model category $B$ has a factorization through $[C^{op}, sSet]_{proj}$ as above, and the category of such factorizations is contractible.

###### Proof

This is theorem 1.1 in

The proof is on page 30.

To produce the factorization $[C^{op},sSet] \to B$ given the functor $\gamma$, first notice that the ordinary Yoneda extension $[C^{op},Set] \to B$ would be given by the left Kan extension given by the coend formula

$F \mapsto \int^{c \in C} \gamma(c) \cdot F(c) \,,$

where the dot in the integrand is the tensoring of cocomplete category $B$ over Set. To refine this to a left Quillen functor $L : [C^{op},sSet] \to B$, choose a cosimplicial resolution?

$\Gamma : C \to [\Delta,B]$

of $\gamma$. Then set

$L : F \mapsto \int^{c \in C} \int^{[n] \in \Delta} \Gamma^n(c) \cdot F_n(c) \,.$

The right adjoint $R : B \to [C^{op},sSet]$ of this functor is given by

$R(X) : c \mapsto Hom_B(\Gamma^\bullet(c), X) \,.$

For $(L \dashv R) : [C^{op}, sSet]_{proj} \stackrel{\to}{\leftarrow} B$ to be a Quillen adjunction, it is sufficient to check that $R$ preserves fibrations and acyclic fibrations. By definition of the projective model structure this means that for every (acyclic) fibration $b_1 \to b_2$ in $B$ we have for every object $c \in C$ that that

$Hom_C(\Gamma^\bullet(c), b_1 \to b_2)$

is an (acyclic) fibration of simplicial sets. But this is one of the standard properties of cosimplicial resolution?s.

Finally, to find the natural weak equivalence $\eta : L \circ j \simeq \gamma$, write $j : C \to [C^{op},sSet]$ for the Yoneda embedding and notice that by Yoneda reduction it follows that for $x \in C$ we have

$L(j(x)) = \int^{c \in C} \int^{[n] \in \Delta} \Gamma^n(c) \cdot C(c,x) = \Gamma^0(x)$

(where equality signs denote isomorphisms).

By the very definition of cosimplicial resolutions, there is a natural weak equivalence $\Gamma(x) \stackrel{\simeq}{\to}$. We can take this to be the component of $\eta$.

###### Corollary

The (∞,1)-Yoneda embedding $j : C \to PSh(C)$ generates $PSh(C)$ under small colimits:

a full (∞,1)-subcategory of $PSh(C)$ that contains all representables and is closed under forming $(\infty,1)$-colimits is already equivalent to $PSh(C)$.

###### Proof

This is HTT, corollary 5.1.5.8.

### Interaction with forming overcategories

The following analog of the corresponding result for 1-categories of presheaves holds for $(\infty,1)$-presheaves. See functors and comma categories.

###### Proposition

(forming overcategories commutes with passing to presheaves)

Let $C$ be a small (∞,1)-category and $p : K \to C$ a diagram. Write $C_{/p}$ and $PSh(C)/_{j p}$ for the corresponding over categories, where $j : C \to PSh(C)$ is the (∞,1)-Yoneda embedding.

Then we have an equivalence of (∞,1)-categories

$PSh(C_{/p}) \stackrel{\simeq}{\to} PSh(C)_{/j p} \,.$

This appears as HTT, 5.1.6.12.

## $(\infty,1)$-subcategories of $(\infty)$-presheaf categories

### Locally presentable $(\infty,1)$-categories

A reflective (∞,1)-subcategory of an $(\infty,1)$-category of $(\infty,1)$-presheaves is called a presentable (∞,1)-category.

### $(\infty,1)$-Sheaf $(\infty,1)$-categories

If that left adjoint (∞,1)-functor to the embedding of the reflective (∞,1)-subcategory furthermore preserves finite limits, then the subcategory is an (∞,1)-category of (∞,1)-sheaves: an (∞,1)-topos

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categoriessatisfying Giraud's axiomsinclusion of left exaxt localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes$\hookrightarrow$algebraic lattices$\simeq$ Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes$\hookrightarrow$locally presentable categories$\simeq$ Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories$\hookrightarrow$accessible categories
model category theorymodel toposes$\hookrightarrow$combinatorial model categories$\simeq$ Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes$\hookrightarrow$locally presentable (∞,1)-categories$\simeq$
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories$\hookrightarrow$accessible (∞,1)-categories

## References

This is the topic of section 5.1 of

Revised on October 15, 2012 18:04:58 by Urs Schreiber (82.113.99.246)