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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 (∞,1)-topoi behave much like the $(\infty,1)$-topos ∞Grpd, their objects are generalized spaces with higher homotopies that may carry more structure. For instance, an ∞-Lie groupoid is an (∞,1)-sheaf on CartSp.

Definition

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

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 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:

• $\mathbf{H}$ 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, theorm 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-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

• 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

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

• 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

  • 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 localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

(n,r)-categories	toposes	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)-topos 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.

An overview is in

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