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 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 $S$ and an accessible left exact? (∞,1)-functor $\overline{(-)}:\mathrm{PSh}(S)\to X$ from (∞,1)-presheaves on $X$, which has a fully faithful right adjoint.

• $X$ 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.

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.

References

Section 6.1 of

• Jacob Lurie, Higher Topos Theory