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(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 categories and sheaves 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 $(\mathrm{\infty }\mathrm{,1})$-topoi as the generalization of the above situation from $1$ to $(\mathrm{\infty }\mathrm{,1})$ (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 topology. The point of ∞-stacks is to pass to parameterized (∞,0)-categories, namely (∞,1)-presheaf categories: these (∞,1)-topoi behave much like the $(\mathrm{\infty }\mathrm{,1})$-category ∞Grpd but their objects are generalized spaces with higher homotopies that may carry more structure, for instance they may be $\mathrm{\infty }$-differentiable stacks if one considers ∞-stacks on Diff.

Definition

A Grothendieck–Rezk–Lurie $(\mathrm{\infty }\mathrm{,1})$-topos is an (∞,1)-category $X$ satisfying the following equivalent conditions:

• $X$ is an (∞,1)-category of (∞,1)-sheaves (meaning: of ∞-stacks):

  there exists a small (∞,1)-category $C$ such that $X$ is a reflective (∞,1)-subcategory $X\stackrel{\stackrel{\mathrm{lex}}{\leftarrow }}{\hookrightarrow }{\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ of the (∞,1)-category of (∞,1)-presheaves ${\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ on $C$.

• $X$ satisfies the $(\mathrm{\infty }\mathrm{,1})$-categorical analogs of Giraud's axioms:

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

The equivalence of these two characterizations is one of the main theorems of HTT.

The second characterization is derived from the following equivalent one:

an (∞,1)-topos is

• a presentable (∞,1)-category

• with universal colimits

• and with object classifiers.

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

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

for the moment see

• topological localization

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

for the moment see

• hypercomplete (∞,1)-topos

Models

Another main theorem about $(\mathrm{\infty }\mathrm{,1})$-toposes is that models for ∞-stack (∞,1)-toposes are given by the model structure on simplicial presheaves. See there for details

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

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

For instance there are evident notions of

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

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

• structures in an (∞,1)-topos.

But $(\mathrm{\infty }\mathrm{,1})$-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.

References

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-Toë-Vezzosi models indeed precisely model $\mathrm{\infty }$-stack $(\mathrm{\infty }\mathrm{,1})$-toposes. Details on this relation are at models for ∞-stack (∞,1)-toposes.