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(infinity,1)-topos

Contents

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

Recall that sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes and that the inclusion functor is necessarily an accessible functor. This characterization has the following immediate generalization to a definition in (∞,1)-category theory, where the only subtlety is that accessibility needs to be explicitly required:

Definition

A GrothendieckRezkLurie (,1)(\infty,1)-topos H\mathbf{H} is an accessible 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 above localization is a topological localization then H\mathbf{H} is an (∞,1)-category of (∞,1)-sheaves.

By Giraud-Rezk-Lurie axioms

Equivalently:

Proposition

(Giraud-Rezk-Lurie axioms)
An (,1)(\infty,1)-topos H\mathbf{H} is

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

This is part of the statement of HTT, theorem 6.1.0.6.

This is derived from the following equivalent one:

Remark

An object classifier is a (small) self-reflection of the \infty-topos inside itself (type of types, internal universe). See also (WdL, book 2, section 1).

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

Proposition

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

Cubical type theory

The Cartesian cubical model of cubical type theory and homotopy type theory is conjectured to be an (∞,1)-topos not equivalent to (∞,1)-groupoids.

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) \;\colon\; \mathbf{H} \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\;\; \bot \;\;\;} \infty Grpd \,.

In fact, this is unique, up to equivalence: Since every \infty-groupoid is an ( , 1 ) (\infty,1) -colimit (namely over itself, by this Prop.) of the point (hence of the terminal object), and since the inverse image \infty-functor LConstLConst needs to preserve these \infty-colimits (being a left adjoint) as well as the point (being a lex functor).

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] \,\colon\, 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) \,.

Powering of \infty-toposes over \infty-groupoids

We discuss how the powering of \infty -toposes over Grpd Grpd_\infty is given by forming mapping stacks out of locally constant \infty -stacks. All of the following formulas and their proogfs hold verbatim also for Grothendieck toposes, as they just use general abstract properties.


Let H\mathbf{H} be an \infty -topos

  • with terminal geometric morphism denoted

    (1)HΓLConstGrp , \mathbf{H} \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} Grp_\infty \,,

    where the inverse image constructs locally constant \infty -stacks,

  • and with its internal hom (mapping stack) adjunction denoted

    (2)HMaps(X,)()×XH \mathbf{H} \underoverset {\underset{Maps(X,-)}{\longrightarrow}} { \overset{ (-) \times X }{\longleftarrow} } {\;\;\;\; \bot \;\;\;\;} \mathbf{H}

    for XHX \,\in\, \mathbf{H}.

    Notice that this construction is also \infty -functorial in the first argument: Maps(XfY,A)Maps\big( X \xrightarrow{f} Y ,\, A \big) is the morphism which under the \infty -Yoneda lemma over H\mathbf{H} (which is large but locally small, so that the lemma does apply) corresponds to

H((),Maps(X,A))H(()×X,A)H(()×f,A)H(()×Y,A)H((),Maps(X,A)). \mathbf{H} \big( (-) ,\, Maps(X,A) \big) \;\simeq\; \mathbf{H} \big( (-) \times X ,\, A \big) \xrightarrow{ \mathbf{H} \big( (-) \times f ,\, A \big) } \mathbf{H} \big( (-) \times Y ,\, A \big) \;\simeq\; \mathbf{H} \big( (-) ,\, Maps(X,A) \big) \,.

By definition, for any SGrpd S \in Grpd_\infty 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

while the tensoring is the (∞,1)-colimit over the diagram constant on XX

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

Remark

Under Isbell duality, the powering operations on homotopy types XX corresponds to higher order Hochschild cohomology of suitable algebras of functions on XX, as discussed there.

Proposition

The powering of H\mathbf{H} over Grpd Grpd_\infty is given by the mapping stack out of the locally constant \infty -stacks:

Grpd op×H LConst op×id H op×H Maps(,) H \array{ Grpd_\infty^{op} \times \mathbf{H} & \overset{ LConst^{op} \times \mathrm{id} }{\longrightarrow} & \mathbf{H}^{op} \times \mathbf{H} & \overset{Maps(-,-)}{\longrightarrow} & \mathbf{H} }

in that this operation has the following properties:

  1. For all X,AHX,\,A \,\in\, \mathbf{H} and SGrpd S \,\in\, Grpd_\infty we have a natural equivalence

    H(X,Maps(LConst(S),A))Grpd (S,H(X,A)) \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) \;\; \simeq \;\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big)
  2. In its first argument the operation

    1. sends the terminal object (the point) to the identity:

      (3)Maps(LConst(*),X)X Maps \big( LConst(\ast) ,\, X \big) \;\; \simeq \;\; X
    2. sends \infty -colimits to \infty -limits:

      (4)Maps(limLConst(S ),X)limMaps(LConst(S ),X), Maps \Big( \underset{ \longrightarrow }{\lim} \, LConst\big(S_\bullet\big) ,\, X \Big) \;\; \simeq \;\; \underset{ \longleftarrow }{\lim} \, Maps \Big( LConst\big(S_\bullet\big) ,\, X \Big) \,,

    where all equivalences shown are natural.

Proof

For the first statement to be proven, consider the following sequence of natural equivalences:

H(X,Maps(LConst(S),A)) H(X×LConst(S),A) (2) H(LConst(S),Maps(X,A)) (2) Grpd (S,ΓMaps(X,A)) (1) Grpd (S,H(* H,Maps(X,A))) bythis Prop. Grpd (S,H(* H×X,A)) (2) Grpd (S,H(X,A)) \begin{array}{lll} \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) & \;\simeq\; \mathbf{H} \big( X \times LConst(S) ,\, A \big) & \text{(2)} \\ & \;\simeq\; \mathbf{H} \Big( LConst(S) ,\, Maps \big( X ,\, A \big) \Big) & \text{(2)} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \Gamma \, Maps \big( X ,\, A \big) \Big) & \text{ (1) } \\ & \;\simeq\; Grpd_\infty \bigg( S ,\, \mathbf{H} \Big( \ast_{\mathbf{H}} ,\, Maps \big( X ,\, A \big) \Big) \bigg) & \text{by}\;\text{<a href="https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject">this Prop.</a>} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( \ast_{\mathbf{H}} \times X ,\, A \big) \Big) & \text{(2)} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big) \end{array}

For the second statement, recall that hom-functors preserve limits in that there are natural equivalences of the form

(5)H(limi,X i,limj,A j)limilimjH(X i,A j), \mathbf{H} \Big( \underset{\underset{i}{\longrightarrow}}{\lim} \,, X_i ,\, \underset{\underset{j}{\longleftarrow}}{\lim} \,, A_j \Big) \;\; \simeq \;\; \underset{\underset{i}{\longleftarrow}}{\lim} \, \underset{\underset{j}{\longleftarrow}}{\lim} \, \mathbf{H} \Big( X_i ,\, A_j \Big) \,,

and that \infty-toposes have universal colimits, in particular that the product operation is a left adjoint (2) and hence preserves colimits:

(6)()×limS lim(()×S ). (-) \,\times\, \underset{{\longrightarrow}}{\lim} \, S_\bullet \;\; \simeq \;\; \underset{{\longrightarrow}}{\lim} \, \big( (-) \,\times\, S_\bullet \big) \,.

With this, we get the following sequences of natural equivalences:

H((),Maps(limLConst(S ),X)) H(()×limLConst(S ),X) (2) H(lim(()×LConst(S )),X) (6) limH(()×LConst(S ),X) (5) limH((),Maps(LConst(S ),X)) (2) H((),limMaps(LConst(S ),X)) (5) . \begin{array}{lll} & \mathbf{H} \bigg( (-) ,\, Maps \Big( \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) \bigg) \\ & \;\simeq\; \mathbf{H} \Big( (-) \times \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) & \text{ (2) } \\ & \;\simeq\; \mathbf{H} \Big( \underset{\longrightarrow}{\lim} \big( (-) \times LConst(S_\bullet) \big) ,\, X \Big) & \text{ (6) } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \big( (-) \times LConst(S_\bullet) ,\, X \big) & \text{ (5) } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \Big( (-) ,\, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ (2) } \\ & \;\simeq\; \mathbf{H} \Big( (-) ,\, \underset{\longleftarrow}{\lim} \, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ (5) } \,. \end{array}

This implies (4) by the \infty -Yoneda lemma over H\mathbf{H} (which is large but locally small, so that the lemma does apply).

Finally (3) is immediate from the fact that LConstLConst preserves the terminal object, by definition:

Maps(LConst(*),X)Maps(* H,X)X. Maps \big( LConst(\ast) ,\, X \big) \;\simeq\; Maps \big( \ast_{\mathbf{H}} ,\, X \big) \;\simeq\; X \,.

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

Proposition

For H\mathbf{H} an (,1)(\infty,1)-topos and XHX \in \mathbf{H} an object, the slice-(∞,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} \xrightarrow{\simeq} \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.

flavors of higher toposes

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

A\phantom{A}(n,r)-categoriesA\phantom{A}A\phantom{A}toposesA\phantom{A}locally presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický‘s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger's theoremglobal model structures on simplicial presheavesn/a
(∞,1)-category theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories

References

General

In retrospect, at least the homotopy categories of (∞,1)-toposes have been known since

presented there via categories of fibrant objects among simplicial presheaves. The enhancement of this to model categories of simplicial presheaves originates wit:h

A more intrinsic characterization of these “model toposes” (Rezk 2010) as \infty-toposes (the term seems to first appear here in Simpson 1999) is due to:

The generalization of these model toposes from 1-sites to simplicial model sites is due to

The term \infty-topos is due to

The term model topos was later coined in:

A comprehensive conceptualization and discussion of (∞,1)-toposes is then due to

building on Rezk 2010. 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.

Overview:

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

A quick introduction to the topic is in

Giraud-Rezk-Lurie 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

Homotopy type theory

Proof that all ∞-stack (∞,1)-topos have presentations by model categories which interpret (provide categorical semantics) for homotopy type theory with univalent type universes:

Last revised on November 25, 2021 at 02:19:22. See the history of this page for a list of all contributions to it.