nLab
(infinity,1)-topos

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

Idea

Recall the following familiar 1-categorical statement:

The idea of (,1)(\infty,1)-toposes is to generalize the above situation from 11 to (,1)(\infty,1) (recall the notion of (n,r)-category and see the general discussion at ∞-topos):

Definition

As a geometric embedding into a (,1)(\infty,1)-presheaf category

Definition

A GrothendieckRezkLurie (,1)(\infty,1)-topos H\mathbf{H} is an accesible left exact reflective sub-(∞,1)-category of an (∞,1)-category of (∞,1)-presheaves

HlexPSh (,1)(C). \mathbf{H} \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,.

If the localization in is a topological localization then H\mathbf{H} is an (∞,1)-category of (∞,1)-sheaves.

By (,1)(\infty,1)-Giraud’s axioms

Equivalently: an (,1)(\infty,1)-topos H\mathbf{H} is

an (∞,1)-category that satisfies the (,1)(\infty,1)-categorical analogs of Giraud's axioms:

The equivalence of these two characterizations is part of the statement of HTT, theorm 6.1.0.6.

This is derived from the following equivalent one:

an (∞,1)-topos is

A further equivalent one (essentially by an invocation of the adjoint functor theorem) is:

an (∞,1)-topos is

Morphisms

A morphism between (,1)(\infty,1)-toposes is an (∞,1)-geometric morphism.

The (∞,1)-category of all (,1)(\infty,1)-topos is (∞,1)Toposes.

Types of (,1)(\infty,1)-toposes

Topological localizations / (,1)(\infty,1)-sheaf toposes

for the moment see

Hypercomplete (,1)(\infty,1)-toposes

for the moment see

Models

Another main theorem about (,1)(\infty,1)-toposes is that models for ∞-stack (∞,1)-toposes are given by the model structure on simplicial presheaves. See there for details

Properties

Global sections geometric morphism

Every ∞-stack (,1)(\infty,1)-topos H\mathbf{H} has a canonical (∞,1)-geometric morphism to the terminal \infty-stack (,1)(\infty,1)-topos ∞Grpd: the direct image is the global sections (∞,1)-functor Γ\Gamma, the inverse image is the constant ∞-stack functor

(LConstΓ):HΓLConstGrpd. (LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,.

Powering and copowering over Grpd\infty Grpd – Hochschild homology

Being a locally presentable (∞,1)-category, an (,1)(\infty,1)-topos H\mathbf{H} is powered and copowered over ∞Grpd, as described at (∞,1)-tensoring.

For any KGrpdK \in \infty Grpd and XHX \in \mathbf{H} the powering is the (∞,1)-limit over the diagram constant on XX

X K=lim KX X^K = {\lim_\leftarrow}_K X

and the (,1)(\infty,1)-copowering is is the (∞,1)-colimit over the diagram constant on XX

KX=lim KX. K \cdot X = {\lim_{\to}}_K X \,.

Under Isbell duality the powering operation corresponds to higher order Hochschild cohomology in XX, as discussed there.

Below we discuss that the powering is equivalently given by the internal hom (mapping stack) out of the constant ∞-stack LConstKLConst K on KK:

X K[LConstK,X]. X^K \simeq [LConst K, X] \,.

Closed monoidal structure

Proposition

Every (,1)(\infty,1)-topos is a cartesian closed (∞,1)-category.

Proof

By the fact that every (,1)(\infty,1)-topos H\mathbf{H} has universal colimits it follows that for every object XX the (∞,1)-functor

X×():HH X \times (-) : \mathbf{H} \to \mathbf{H}

preserves all (∞,1)-colimits. Since every (,1)(\infty,1)-topos is a locally presentable (∞,1)-category it follows with the adjoint (∞,1)-functor theorem that there is a right adjoint (∞,1)-functor

(X×()[X,]):H[X,]X×()H. (X \times (-) \dashv [X,-]) : \mathbf{H} \stackrel{\overset{X \times (-)}{\leftarrow}}{\underset{[X,-]}{\to}} \mathbf{H} \,.
Proposition

For CC an (∞,1)-site for H\mathbf{H} we have that the internal hom (mapping stack) [X,][X,-] is given on AHA \in \mathbf{H} by the (∞,1)-sheaf

[X,A]:UH(X×Ly(U),A), [X,A] : U \mapsto \mathbf{H}(X \times L y(U), A) \,,

where y:CHy : C \to \mathbf{H} is the (∞,1)-Yoneda embedding and L:PSh CHL : PSh_C \to \mathbf{H} denotes ∞-stackification.

Proof

The argument is entirely analogous to that of the closed monoidal structure on sheaves.

We use the full and faithful geometric embedding (Li):HPSh C(L \dashv i) : \mathbf{H} \hookrightarrow PSh_C and the (∞,1)-Yoneda lemma to find for all UCU \in C the value

[X,A](U)PSh C(yU,[X,A]) [X,A](U) \simeq PSh_C(y U, [X,A])

and then the fact that ∞-stackification LL is left adjoint to inclusion to get

H(Ly(U),[X,A]). \cdots \simeq \mathbf{H}(L y(U), [X,A]) \,.

Then the defining adjunction (X×()[X,])(X \times (-) \dashv [X,-]) gives

H(X×Ly(U),A). \cdots \simeq \mathbf{H}(X \times L y(U) , A) \,.
Proposition

Finite colimits may be taken out of the internal hom: For II a finite (,1)(\infty,1)-category and X:IHX : I \to \mathbf{H} a diagram, we have for all AHA \in \mathbf{H}

[lim iX i,A]lim i[X i,A] [{\lim_\to}_i X_i, A] \simeq {\lim_\leftarrow}_i [X_i,A]
Proof

By the above proposition we have

[lim iX i,A](U)H((lim iX i)×Ly(U),A). [{\lim_\to}_i X_i, A](U) \simeq \mathbf{H}(({\lim_\to}_i X_i) \times L y(U), A) \,.

By universal colimits in H\mathbf{H} this is

H(lim iX i×Ly(U),A). \cdots \simeq \mathbf{H}({\lim_\to}_i X_i \times L y(U), A) \,.

Using the fact that the hom-functor sends colimits in the first argument to limits this is

lim iH(X i×LyU,A). \cdots \simeq {\lim_\leftarrow}_i \mathbf{H}(X_i \times L y U, A) \,.

By the internal hom adjunction and Yoneda this is

lim i[X i,A](U). \cdots \simeq {\lim_\leftarrow}_i [X_i, A](U) \,.

Since (∞,1)-limits in the (∞,1)-category of (∞,1)-presheaves are computed objectwise, this is

(lim i[X i,A])(U). \cdots \simeq ({\lim_\leftarrow}_i [X_i,A])(U) \,.

Finally, because LL is a left exact (∞,1)-functor this is also the (∞,1)-limit in H\mathbf{H}.

Proposition

For SS \in ∞Grpd write LConstSLConst S for its inverse image under the global section (∞,1)-geometric morphism (LConstΓ):HGrpd(LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd: the constant ∞-stack on SS.

Then the internal hom [LConstS,A][LConst S,A] coincides with the (∞,1)-powering of AA by SS:

[LConstS,A]A S [LConst S, A] \simeq A^S
Proof

By the above we have

[LConstS,A](U)H(LConstS×Ly(U),A). [LConst S, A](U) \simeq \mathbf{H}(LConst S \times L y(U), A) \,.

As the notation indicates, LConstSLConst S is precisely LConstSL Const S: the ∞-stackification of the (∞,1)-presheaf that is literally constant on SS. Morover LL is a left exact (∞,1)-functor and hence commutes with (∞,1)-products, so that

H(L(ConstS×y(U)),A). \cdots \simeq \mathbf{H}(L(Const S \times y(U)), A) \,.

By the defining geometric embedding (Li)(L \dashv i) this is

PSh C(ConstS×y(U),A). \cdots \simeq PSh_C(Const S \times y(U), A) \,.

Since limits of (∞,1)-presheaves are taken objectwise, we have in the first argument the tensoring of y(U)y(U) over SS

PSh C(Sy(U),A). \cdots \simeq PSh_C(S \cdot y(U), A) \,.

By the defining property of tensoring and cotensoring (or explicitly writing out Sy(U)=lim Sconsty(U)S \cdot y(U) = {\lim_\to}_{S} const y(U) , taking the colimit out of the hom, thus turning it into a limit and then inserting that back in the second argument) this is

PSh C(y(U),A S). \cdots \simeq PSh_C(y(U), A^S) \,.

So finally with the (∞,1)-Yoneda lemma we have

A S(U). \cdots \simeq A^S(U) \,.

Over-(,1)(\infty,1)-toposes

Proposition

For H\mathbf{H} an (,1)(\infty,1)-topos and XHX \in \mathbf{H} an object, the over-(∞,1)-category H /X\mathbf{H}_{/X} is itself an (,1)(\infty,1)-topos – an over-(∞,1)-topos. The projection π !:H /XH\pi_! : \mathbf{H}_{/X} \to \mathbf{H} part of an essential geometric morphism

π:H /Xπ *π *π !H. \pi : \mathbf{H}_{/X} \stackrel{\overset{\pi_!}{\to}}{\stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}}} \mathbf{H} \,.

This is HTT, prop. 6.3.5.1.

The (,1)(\infty,1)-topos H /X\mathbf{H}_{/X} could be called the gros topos of XX. A geometric morphism KH\mathbf{K} \to \mathbf{H} that factors as KH /XπH\mathbf{K} \to \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H} is called an etale geometric morphism.

Syntax in univalent homotopy type theory

(,1)(\infty,1)-Toposes provide categorical semantics for homotopy type theory with a univalent Tarskian type of types (which inteprets as the object classifier).

For more on this see at

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

Most of the standard constructions in topos theory have or should have immediate generalizations to the context of (,1)(\infty,1)-toposes, since all notions of category theory exist for (∞,1)-categories.

For instance there are evident notions of

Moreover, it turns out that (,1)(\infty,1)-toposes come with plenty of internal structures, more than canonically present in an ordinary topos. Every (,1)(\infty,1)-topos comes with its intrinsic notion of

and with an intrinsic notion of

In classical topos theory, cohomology and homotopy of a topos EE are defined in terms of simplicial objects in CC. If EE is a sheaf topos with site CC and enough points, then this classical construction is secretly really a model for the intrinsic cohomology and homotopy in the above sense of the hypercomplete (∞,1)-topos of ∞-stacks on CC.

The beginning of a list of all the structures that exist intrinsically in a big (,1)(\infty,1)-topos is at

But (,1)(\infty,1)-topos theory in the style of an \infty-analog of the Elephant is only barely beginning to be conceived.

There are some indications as to what the

should be.

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

General

In retrospect it turns out that the homotopy categories of (∞,1)-toposes have been known since

And the model category theory models have been known since Andre Joyal proposed the model structure on simplicial sheaves in his letter to Alexander Grothendieck.

This work used 1-categorical sites. The generalization to (∞,1)categorical sites – modeled by model sites – was discussed in

and “model topos”-theory was also developed in

The intrinsic category-theoretic definition of an (∞,1)-topos was given in section 6.1 of

building on ideas by Charles Rezk. There is is also proven that the Brown-Joyal-Jardine-Toën-Vezzosi models indeed precisely model \infty-stack (,1)(\infty,1)-toposes. Details on this relation are at models for ∞-stack (∞,1)-toposes.

A useful collection of facts of simplicial homotopy theory and (infinity,1)-topos theory is in

\infty-Giraud axioms

A discussion of the (,1)(\infty,1)-universal colimits in terms of model category presentations is due to

  • Charles Rezk, Fibrations and homotopy colimits of simplicial sheaves (pdf)

More on this with an eye on associated ∞-bundles is in

Revised on June 26, 2014 08:30:36 by Urs Schreiber (109.246.71.66)