# nLab (infinity,1)-topos

### Context

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

(∞,1)-category theory

## Models

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

Recall the following familiar 1-categorical statement:

One can think of $\left(\infty ,1\right)$-toposes as the generalization of the above situation from $1$ to $\left(\infty ,1\right)$ (recall the notion of (n,r)-category and see the general discussion at ∞-topos):

## Definition

### As a geometric embedding into a $\left(\infty ,1\right)$-presheaf category

###### Definition

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

$H\stackrel{\stackrel{\mathrm{lex}}{←}}{↪}{\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H} \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,.

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

### By $\left(\infty ,1\right)$-Giraud’s axioms

Equivalently: an $\left(\infty ,1\right)$-topos $H$ is

an (∞,1)-category that satisfies the $\left(\infty ,1\right)$-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

### Morphism

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

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

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

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

for the moment see

### Hypercomplete $\left(\infty ,1\right)$-toposes

for the moment see

## Models

Another main theorem about $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-topos $H$ has a canonical (∞,1)-geometric morphism to the terminal $\infty$-stack $\left(\infty ,1\right)$-topos ∞Grpd: the direct image is the global sections (∞,1)-functor $\Gamma$, the inverse image is the constant ∞-stack functor

$\left(\mathrm{LConst}⊣\Gamma \right):H\stackrel{\stackrel{\mathrm{LConst}}{←}}{\underset{\Gamma }{\to }}\infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$(LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,.

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

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

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

${X}^{K}={\underset{←}{\mathrm{lim}}}_{K}X$X^K = {\lim_\leftarrow}_K X

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

$K\cdot X={\underset{\to }{\mathrm{lim}}}_{K}X\phantom{\rule{thinmathspace}{0ex}}.$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 $\mathrm{LConst}K$ on $K$:

${X}^{K}\simeq \left[\mathrm{LConst}K,X\right]\phantom{\rule{thinmathspace}{0ex}}.$X^K \simeq [LConst K, X] \,.

### Closed monoidal structure

###### Proposition

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

###### Proof

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

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

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

$\left(X×\left(-\right)⊣\left[X,-\right]\right):H\stackrel{\stackrel{X×\left(-\right)}{←}}{\underset{\left[X,-\right]}{\to }}H\phantom{\rule{thinmathspace}{0ex}}.$(X \times (-) \dashv [X,-]) : \mathbf{H} \stackrel{\overset{X \times (-)}{\leftarrow}}{\underset{[X,-]}{\to}} \mathbf{H} \,.
###### Proposition

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

$\left[X,A\right]:U↦H\left(X×Ly\left(U\right),A\right)\phantom{\rule{thinmathspace}{0ex}},$[X,A] : U \mapsto \mathbf{H}(X \times L y(U), A) \,,

where $y:C\to H$ is the (∞,1)-Yoneda embedding and $L:{\mathrm{PSh}}_{C}\to H$ 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 $\left(L⊣i\right):H↪{\mathrm{PSh}}_{C}$ and the (∞,1)-Yoneda lemma to find for all $U\in C$ the value

$\left[X,A\right]\left(U\right)\simeq {\mathrm{PSh}}_{C}\left(yU,\left[X,A\right]\right)$[X,A](U) \simeq PSh_C(y U, [X,A])

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

$\cdots \simeq H\left(Ly\left(U\right),\left[X,A\right]\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq \mathbf{H}(L y(U), [X,A]) \,.

Then the defining adjunction $\left(X×\left(-\right)⊣\left[X,-\right]\right)$ gives

$\cdots \simeq H\left(X×Ly\left(U\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(\infty ,1\right)$-category and $X:I\to H$ a diagram, we have for all $A\in H$

$\left[{\underset{\to }{\mathrm{lim}}}_{i}{X}_{i},A\right]\simeq {\underset{←}{\mathrm{lim}}}_{i}\left[{X}_{i},A\right]$[{\lim_\to}_i X_i, A] \simeq {\lim_\leftarrow}_i [X_i,A]
###### Proof

By the above proposition we have

$\left[{\underset{\to }{\mathrm{lim}}}_{i}{X}_{i},A\right]\left(U\right)\simeq H\left(\left({\underset{\to }{\mathrm{lim}}}_{i}{X}_{i}\right)×Ly\left(U\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$[{\lim_\to}_i X_i, A](U) \simeq \mathbf{H}(({\lim_\to}_i X_i) \times L y(U), A) \,.

By universal colimits in $H$ this is

$\cdots \simeq H\left({\underset{\to }{\mathrm{lim}}}_{i}{X}_{i}×Ly\left(U\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\cdots \simeq {\underset{←}{\mathrm{lim}}}_{i}H\left({X}_{i}×LyU,A\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq {\lim_\leftarrow}_i \mathbf{H}(X_i \times L y U, A) \,.

By the internal hom adjunction and Yoneda this is

$\cdots \simeq {\underset{←}{\mathrm{lim}}}_{i}\left[{X}_{i},A\right]\left(U\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq {\lim_\leftarrow}_i [X_i, A](U) \,.

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

$\cdots \simeq \left({\underset{←}{\mathrm{lim}}}_{i}\left[{X}_{i},A\right]\right)\left(U\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $H$.

###### Proposition

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

Then the internal hom $\left[\mathrm{LConst}S,A\right]$ coincides with the (∞,1)-powering of $A$ by $S$:

$\left[\mathrm{LConst}S,A\right]\simeq {A}^{S}$[LConst S, A] \simeq A^S
###### Proof

By the above we have

$\left[\mathrm{LConst}S,A\right]\left(U\right)\simeq H\left(\mathrm{LConst}S×Ly\left(U\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$[LConst S, A](U) \simeq \mathbf{H}(LConst S \times L y(U), A) \,.

As the notation indicates, $\mathrm{LConst}S$ is precisely $L\mathrm{Const}S$: 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

$\cdots \simeq H\left(L\left(\mathrm{Const}S×y\left(U\right)\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq \mathbf{H}(L(Const S \times y(U)), A) \,.

By the defining geometric embedding $\left(L⊣i\right)$ this is

$\cdots \simeq {\mathrm{PSh}}_{C}\left(\mathrm{Const}S×y\left(U\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $y\left(U\right)$ over $S$

$\cdots \simeq {\mathrm{PSh}}_{C}\left(S\cdot y\left(U\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq PSh_C(S \cdot y(U), A) \,.

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

$\cdots \simeq {\mathrm{PSh}}_{C}\left(y\left(U\right),{A}^{S}\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq PSh_C(y(U), A^S) \,.

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

$\cdots \simeq {A}^{S}\left(U\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq A^S(U) \,.

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

###### Proposition

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

$\pi :{H}_{/X}\stackrel{\stackrel{{\pi }_{!}}{\to }}{\stackrel{\stackrel{{\pi }^{*}}{←}}{\underset{{\pi }_{*}}{\to }}}H\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(\infty ,1\right)$-topos ${H}_{/X}$ could be called the gros topos of $X$. A geometric morphism $K\to H$ that factors as $K\to {H}_{/X}\stackrel{\pi }{\to }H$ is called an etale geometric morphism.

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

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

For instance there are evident notions of

Moreover, it turns out that $\left(\infty ,1\right)$-toposes come with plenty of internal structures, more than canonically present in an ordinary topos. Every $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-topos is at

But $\left(\infty ,1\right)$-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$↪$algebraic lattices$\simeq$ Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes$↪$locally presentable categories$\simeq$ Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories$↪$accessible categories
model category theorymodel toposes$↪$combinatorial model categories$\simeq$ Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes$↪$locally presentable (∞,1)-categories$\simeq$
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories$↪$accessible (∞,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 – modeld 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 $\left(\infty ,1\right)$-toposes. Details on this relation are at models for ∞-stack (∞,1)-toposes.

### $\infty$-Giraud axioms

A discussion of the $\left(\infty ,1\right)$-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 February 23, 2013 07:52:59 by Mike Shulman (192.16.204.218)