# nLab (infinity,1)-topos

Contents

### Context

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

(∞,1)-category theory

## Models

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

Recall the following familiar 1-categorical statement:

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

## Definition

### As a geometric embedding into a $(\infty,1)$-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 $(\infty,1)$-topos $\mathbf{H}$ is an accessible left exact reflective sub-(∞,1)-category of an (∞,1)-category of (∞,1)-presheaves

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

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

### By Giraud-Rezk-Lurie axioms

Equivalently:

###### Proposition

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

an (∞,1)-category that satisfies the $(\infty,1)$-categorical 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:

###### Proposition

An (∞,1)-topos is

###### Remark

An object classifier is a (small) self-reflections 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 $(\infty,1)$-toposes is an (∞,1)-geometric morphism.

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

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

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

for the moment see

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

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

### Powering and copowering over $\infty Grpd$ – Hochschild homology

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

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

$X^K = {\lim_\leftarrow}_K X$

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

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

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

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

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

### Closed monoidal structure

###### Proposition

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

###### Proof

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

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

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

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

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

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

where $y : C \to \mathbf{H}$ is the (∞,1)-Yoneda embedding and $L : PSh_C \to \mathbf{H}$ 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 $(L \dashv i) : \mathbf{H} \hookrightarrow PSh_C$ and the (∞,1)-Yoneda lemma to find for all $U \in C$ the value

$[X,A](U) \simeq PSh_C(y U, [X,A])$

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

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

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

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

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

$[{\lim_\to}_i X_i, A] \simeq {\lim_\leftarrow}_i [X_i,A]$
###### Proof

By the above proposition we have

$[{\lim_\to}_i X_i, A](U) \simeq \mathbf{H}(({\lim_\to}_i X_i) \times L y(U), A) \,.$

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

$\cdots \simeq \mathbf{H}({\lim_\to}_i X_i \times L y(U), A) \,.$

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

$\cdots \simeq {\lim_\leftarrow}_i \mathbf{H}(X_i \times L y U, A) \,.$

By the internal hom adjunction and Yoneda this is

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

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

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

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

###### Proposition

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

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

$[LConst S, A] \simeq A^S$
###### Proof

By the above we have

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

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

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

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

$\cdots \simeq PSh_C(Const S \times y(U), A) \,.$

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

$\cdots \simeq PSh_C(S \cdot y(U), A) \,.$

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

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

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

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

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

###### Proposition

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

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

This is HTT, prop. 6.3.5.1.

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

### Syntax in univalent homotopy type theory

$(\infty,1)$-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

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

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

For instance there are evident notions of

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

and with an intrinsic notion of

In classical topos theory, cohomology and homotopy of a topos $E$ are defined in terms of simplicial objects in $C$. If $E$ is a sheaf topos with site $C$ 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 $C$.

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

But $(\infty,1)$-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: 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.

$\phantom{A}$(n,r)-categories$\phantom{A}$$\phantom{A}$toposes$\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

### 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 $(\infty,1)$-toposes. Details on this relation are at models for ∞-stack (∞,1)-toposes.

An overview is in

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 $(\infty,1)$-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: