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

Context

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

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

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

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

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

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

Properties

Models

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

Proposition

For CC a simplicially enriched category with Kan complexes as hom-objects, write [C op,sSet Quillen] proj[C^{op}, sSet_{Quillen}]_{proj} and [C op,sSet Quillen] inj[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()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))N([C op,sSet Quillen] proj) . 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 (,1)(\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 (,1)(\infty,1)-category of (,1)(\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 CC a small (,1)(\infty,1)-category, the (,1)(\infty,1)-category PSh(C)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 CC is the free cocompletion of CC, the free completion under forming colimits.

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

Lemma

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

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

Proof

This is HTT, lemma 5.1.5.3.

For DD a quasi-category with all small colimits, write Func L(PSh(C),D)Func(PSh(C),D)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:CPSh(C)j : C \to PSh(C) induces an equivalence of quasi-categories

Func L(PSh(C),D)Func(C,D). 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 AA and BB be model categories, DD a plain category and

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

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

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

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

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

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

Lr(d) Lr(d) η d η d γ() \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 LL presents, via its derived functor, a left adjoint (∞,1)-functor that preserves (,1)(\infty,1)-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving (,1)(\infty,1)-functors into (,1)(\infty,1)-categories that have all colimits.

Theorem

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

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

every functor CBC \to B to any model category BB has a factorization through [C op,sSet] proj[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]B[C^{op},sSet] \to B given the functor γ\gamma, first notice that the ordinary Yoneda extension [C op,Set]B[C^{op},Set] \to B would be given by the left Kan extension given by the coend formula

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

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

Γ:C[Δ,B] \Gamma : C \to [\Delta,B]

of γ\gamma. Then set

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

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

R(X):cHom B(Γ (c),X). R(X) : c \mapsto Hom_B(\Gamma^\bullet(c), X) \,.

For (LR):[C op,sSet] projB(L \dashv R) : [C^{op}, sSet]_{proj} \stackrel{\to}{\leftarrow} B to be a Quillen adjunction, it is sufficient to check that RR preserves fibrations and acyclic fibrations. By definition of the projective model structure this means that for every (acyclic) fibration b 1b 2b_1 \to b_2 in BB we have for every object cCc \in C that that

Hom C(Γ (c),b 1b 2) 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 η:Ljγ\eta : L \circ j \simeq \gamma, write j:C[C op,sSet]j : C \to [C^{op},sSet] for the Yoneda embedding and notice that by Yoneda reduction it follows that for xCx \in C we have

L(j(x))= cC [n]ΔΓ n(c)C(c,x)=Γ 0(x) 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 Γ(x)\Gamma(x) \stackrel{\simeq}{\to}. We can take this to be the component of η\eta.

Corollary

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

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

Proposition

(forming overcategories commutes with passing to presheaves)

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

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

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

This appears as HTT, 5.1.6.12.

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

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

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

(,1)(\infty,1)-Sheaf (,1)(\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\hookrightarrowalgebraic lattices\simeq Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes\hookrightarrowlocally presentable categories\simeq Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories\hookrightarrowaccessible categories
model category theorymodel toposes\hookrightarrowcombinatorial model categories\simeq Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes\hookrightarrowlocally presentable (∞,1)-categories\simeq
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories\hookrightarrowaccessible (∞,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)