nLab
(infinity,1)-topos

Idea

Recall the following familiar 1-categorical statement:

• Working in the 1-category Set of 0-categories is the same as doing set theory. The point of sheaf toposes is to pass to parameterized 0-categories, namely presheaf categories: these topoi behave much like the category Set but their objects are generalized spaces that may carry more structure, for instance they may be generalized smooth spaces if one considers (pre)sheaves on Diff.

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

• Working in the (∞,1)-category ∞Grpd of (∞,0)-categories is the same as doing homotopy theory. The point of (∞,1)-sheaves is to pass to parameterized (∞,0)-categories, namely (∞,1)-presheaf categories: these (∞,1)-topoi behave much like the $(\mathrm{\infty },1)$-category ∞Grpd but their objects are generalized spaces with higher homotopies that may carry more structure, for instance they may be ∞-Lie groupoids if one considers (∞,1)-sheaves on CartSp.

Definition

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

A Grothendieck–Rezk–Lurie $(\mathrm{\infty },1)$-topos $H$ is an accessibly embedded reflective sub-(∞,1)-category of an (∞,1)-category of (∞,1)-presheaves

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

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

By $(\mathrm{\infty },1)$-Giraud’s axioms

Equivalently: an $(\mathrm{\infty },1)$-topos $H$ is

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

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

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 presentable (∞,1)-category

• with universal colimits

• and with object classifiers.

Morphism

A morphism between $(\mathrm{\infty },1)$-toposes is an (∞,1)-geometric morphism.

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

Types of $(\mathrm{\infty },1)$-toposes

Topological localizations / $(\mathrm{\infty },1)$-sheaf toposes

for the moment see

• topological localization

Hypercomplete $(\mathrm{\infty },1)$-toposes

for the moment see

• hypercomplete (∞,1)-topos

Models

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

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

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

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

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

$$X^K={\underleftarrow{\lim }}_{K}X$$X^K = {\lim_\leftarrow}_K X

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

$$K\cdot X={\underrightarrow{\lim }}_{K}X.$$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 out of the constant ∞-stack $\mathrm{LConst}K$ on $K$:

$$X^K\simeq [\mathrm{LConst}K,X].$$X^K \simeq [LConst K, X] \,.

Closed monoidal structure

Proposition

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

Proof

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

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

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

$$(X\times (-)⊣[X,-]):H\stackrel{\stackrel{X\times (-)}{\leftarrow }}{\underset{[X,-]}{\to }}H.$$(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 $[X,-]$ is given on $A\in H$ by the (∞,1)-sheaf

$$[X,A]:U\mapsto H(X\times Ly(U),A),$$[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 $(L⊣i):H\hookrightarrow {\mathrm{PSh}}_C$ and the (∞,1)-Yoneda lemma to find for all $U\in C$ the value

$$[X,A](U)\simeq {\mathrm{PSh}}_C(yU,[X,A])$$[X,A](U) \simeq PSh_C(y U, [X,A])

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

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

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

$$\cdots \simeq H(X\times Ly(U),A).$$\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 $(\mathrm{\infty },1)$-category and $X:I\to H$ a diagram, we have for all $A\in H$

$$[{\underrightarrow{\lim }}_i{X}_i,A]\simeq {\underleftarrow{\lim }}_i[X_i,A]$$[{\lim_\to}_i X_i, A] \simeq {\lim_\leftarrow}_i [X_i,A]

Proof

By the above proposition we have

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

By universal colimits in $H$ this is

$$\cdots \simeq H({\underrightarrow{\lim }}_i{X}_i\times Ly(U),A).$$\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

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

By the internal hom adjunction and Yoneda this is

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

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

$$\cdots \simeq ({\underleftarrow{\lim }}_i[X_i,A])(U).$$\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 $(\mathrm{LConst}⊣\Gamma ):H\to \mathrm{\infty }\mathrm{Grpd}$: the constant ∞-stack on $S$.

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

$$[\mathrm{LConst}S,A]\simeq A^S$$[LConst S, A] \simeq A^S

Proof

By the above we have

$$[\mathrm{LConst}S,A](U)\simeq H(\mathrm{LConst}S\times Ly(U),A).$$[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

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

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

$$\cdots \simeq {\mathrm{PSh}}_C(\mathrm{Const}S\times y(U),A).$$\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 $y(U)$ over $S$

$$\cdots \simeq {\mathrm{PSh}}_C(S\cdot y(U),A).$$\cdots \simeq PSh_C(S \cdot y(U), A) \,.

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

$$\cdots \simeq {\mathrm{PSh}}_C(y(U),A^S).$$\cdots \simeq PSh_C(y(U), A^S) \,.

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

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

Over-$(\mathrm{\infty },1)$-toposes

Proposition

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

$$\pi :H_{/X}\stackrel{\stackrel{{\pi }_{!}}{\to }}{\stackrel{\stackrel{{\pi }^{*}}{\leftarrow }}{\underset{{\pi }_{*}}{\to }}}H.$$\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 $(\mathrm{\infty },1)$-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.

$(\mathrm{\infty },1)$-Topos theory

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

For instance there are evident notions of

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

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

• cohesive (∞,1)-topos.

But $(\mathrm{\infty },1)$-topos theory in the style of an $\mathrm{\infty }$-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

• $(\mathrm{\infty },1)$-topos, elementary (∞,1)-topos

• (∞,2)-topos

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 – modeld by model sites – was discussed in

• Bertrand Toën, Gabriele Vezzosi, Homotopical Algebraic Geometry I: Topos theory (arXiv)

and “model topos”-theory was also developed in

• Charles Rezk, Toposes and homotopy toposes (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 $\mathrm{\infty }$-stack $(\mathrm{\infty },1)$-toposes. Details on this relation are at models for ∞-stack (∞,1)-toposes.

$\mathrm{\infty }$-Giraud axioms

A discussion of the $(\mathrm{\infty },1)$-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)