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(infinity,1)-topos

Idea

Recall the following familiar 1-categorical statement:

• Working in the 1-category Set of 0-categories amounts to doing set theory. The point of sheaf toposes is to pass to parameterized 0-categories, namely presheaf categories. Although these topoi behave much like the 1-topos Set, their objects are generalized spaces that may carry more structure. For instance, a (pre)sheaf on Diff is a generalized smooth space.

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):

• Working in the (∞,1)-category ∞Grpd of (∞,0)-categories amounts to doing homotopy theory. The point of (∞,1)-sheaves is to pass to parameterized (∞,0)-categories, namely (∞,1)-presheaf categories. Although these $(\infty,1)$-topoi behave much like the $(\infty,1)$-topos ∞Grpd, their objects are generalized spaces with higher homotopies that may carry more structure. More generally we have topoi of sheaves, and $(\infty,1)$-topoi of (∞,1)-sheaves. For instance, an ∞-Lie groupoid is an (∞,1)-sheaf on CartSp.

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 Grothendieck–Rezk–Lurie $(\infty,1)$-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 above localization 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:

• $\mathbf{H}$ is presentable;
• (∞,1)-colimits in $\mathbf{H}$ are universal;
• coproducts in $\mathbf{H}$ are disjoint;
• every groupoid object in $\mathbf{H}$ is effective (i.e. has a delooping).

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

• a presentable (∞,1)-category with universal colimits

• that has object classifiers.

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

• a presentable (∞,1)-category

• in which all colimits are van Kampen colimits.

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

• topological localization

Hypercomplete $(\infty,1)$-toposes

for the moment see

• hypercomplete (∞,1)-topos

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 $\mathbf{K} \xrightarrow{\simeq} \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}$ 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

• model of type theory in an (infinity,1)-topos

• relation between type theory and category theory – Univalent homotopy type theory and infinity-toposes

$(\infty,1)$-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

• geometric morphisms between $(\infty,1)$-toposes, such as the global section geometric morphism to the terminal (∞,1)-sheaf $(\infty,1)$-topos ∞Grpd.

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

• cohomology in an (∞,1)-topos

and with an intrinsic notion of

• homotopy in an (∞,1)-topos.

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

• cohesive (∞,1)-topos.

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

• internal logic of an (∞,1)-topos

should be.

Related concepts

• (0,1)-topos

• topos

• 2-topos

• $(\infty,1)$-topos, elementary (∞,1)-topos, (∞,1)-pretopos

  • model topos

  • (n,1)-topos

  • structured (∞,1)-topos

  • compact topos, coherent (∞,1)-topos

• category object in an (∞,1)-topos

• (∞,2)-topos

• (∞,n)-topos

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 presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Adámek-Rosický‘s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,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

• Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology .

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

• Bertrand Toën, Gabriele Vezzosi, Homotopical Algebraic Geometry I: Topos theory, Advances in Mathematics 193.2 (2005): 257-372. (arXiv:math.AG/0207028)

and “model topos”-theory was also developed in

• Charles Rezk, Toposes and homotopy toposes, 2010 (pdf)

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

• Jacob Lurie, Higher Topos Theory

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.

Overview

• Denis-Charles Cisinski, Catégories supérieures et théorie des topos, Séminaire Bourbaki, 21.3.2015, pdf.

• Charles Rezk, Lectures on Higher Topos Theory, Leeds 2019 (pdf, pdf)

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

• Zhen Lin Low, Notes on homotopical algebra

A quick introduction to the topic is in

• André Joyal, A crash course in topos theory – The big picture, lecture series at Topos à l’IHES, November 2015, Paris

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

• Matthias Wendt, Classifying spaces and fibrations of simplicial sheaves (arXiv)

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:

• Michael Shulman, All $(\infty,1)$-toposes have strict univalent universes (arXiv:1904.07004).