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

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

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos



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):


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:


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



An (,1)(\infty,1)-topos H\mathbf{H} is

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

This is part of the statement of HTT, theorem

This is derived from the following equivalent one:


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:


An (∞,1)-topos is


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.


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


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).

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


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


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} \,.

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.


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) \,.

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]

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}.


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

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) \,.



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.

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



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 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.


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 October 14, 2021 at 04:24:45. See the history of this page for a list of all contributions to it.