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(infinity,1)-category of (infinity,1)-sheaves

This entry is about the general properties and characterization of (∞,1)-categories of (∞,1)-sheaves – also known as ∞-stack (∞,1)-toposes.

The 1-categorical analog of the discussion is this entry is at category of sheaves.

Idea

For any (∞,1)-category $C$ every (∞,1)-functor $\overline{(-)}:C\to D$ which admits a fully faithful right adjoint – equivalently every (∞,1)-functor $L:C\to C$ which is left adjoint to the inclusion $LC\hookrightarrow C$ of its essential image $LC$ into $C$ – is a localization of an (∞,1)-category onto a reflective (∞,1)-subcategory characterized by the collection $W$ of morphisms which it sends to equivalences?. One can think of it

• as inverting these morphisms;

• as projecting onto those objects of $C$ which are local with respect to these morphsims: those objects which sees $W$ as a collection of equivalences.

Using the familiar characterization of the category of sheaves in the 1-categorical context, this straightforwardly suggests to characterize $(\mathrm{\infty }\mathrm{,1})$-categories $\mathrm{Sh}(S)$ of $(\mathrm{\infty }\mathrm{,1})$-sheaves – also called (Grothendieck-Rezk-Lurie) (∞,1)-topoi as essential images of left exact? $(\mathrm{\infty }\mathrm{,1})$-functors from (∞,1)-categories of (∞,1)-presheaves

$$\overline{(-)}:\mathrm{PSh}(S)\to \mathrm{Sh}(S)$$\bar{(-)} : PSh(S) \to Sh(S)

(called ∞-stackification, analogous to sheafification)

that have a fully faithful right adjoint forgetful functor

$$\mathrm{Sh}(S)\hookrightarrow \mathrm{PSh}(S).$$Sh(S) \hookrightarrow PSh(S) \,.

Definition

In the strict sense of the word, an (∞,1)-category of $(\mathrm{\infty }\mathrm{,1})$-sheaves is a topological localization of an (∞,1)-category of (∞,1)-presheaves ${\mathrm{PSh}}_{(incfty\mathrm{,1})}(C)$ on some small (∞,1)-category $C$

$${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\hookrightarrow {\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(C).$$Sh_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C) \,.

Notice that by the facts recalled there, every such topological localization corresponds uniquely, up to equivalence, to a choice of Grothendieck topology on $C$: the objects of ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ are those $(\mathrm{\infty }\mathrm{,1})$-presheaves that satisfy descent with respect to Cech covers of covering sieves.

Often such $(\mathrm{\infty }\mathrm{,1})$-sheaves are called ∞-stacks. The counting is

• sheaf = 0-stack

• (2,1)-sheaf = stack

• (3,1)-sheaf = 2-stack

• etc

• $(\mathrm{\infty }\mathrm{,1})$-sheaf = ∞-stack.

Notice also that topological localizations are usually not hypercomplete (∞,1)-toposes. With slight abouse of language, the objects of the hypercompletion may still be called ”$\mathrm{\infty }$-stack”s around here.

Models

The topological localizations of an (∞,1)-category of (∞,1)-presheaves are presented by the left Bousfield localization of the global model structure on simplicial presheaves at the set of Cech covers.

The hypercomplete $(\mathrm{\infty }\mathrm{,1})$-sheaf toposes are presented by the local Joyal-Jardine model structure on simplicial presheaves.

Detailed discussion of this model category presentation is at

• models for (infinity,1)-category of (infinity,1)-sheaves .

References

The study of simplicial presheaves apparently goes back to

• K. Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology

which considers locally Kan simplicial presheaves as a category of fibrant objects.

This was later conceived in terms of a model structure on simplicial presheaves and on simplicial sheaves by Joyal and Jardine. Toë summarizes the situation and emphasizes the interpretation in terms of ∞-stacks living in $(\mathrm{\infty }\mathrm{,1})$-categories for instance in

B. Toën, Higher and derived stacks: a global overview (arXiv) .

The full picture in terms of Grothendieck-$(\mathrm{\infty }\mathrm{,1})$-topoi of $(\mathrm{\infty }\mathrm{,1})$-sheaves is the topic of

J. Lurie, Higher Topos Theory.

• localization $(\mathrm{\infty }\mathrm{,1})$-functors ($(\mathrm{\infty }\mathrm{,1})$-sheafification for the present purpose) are discussed in section 5.2.7;

• local objects ($(\mathrm{\infty }\mathrm{,1})$-sheaves for the present purpose) and local isomorphisms are discussed in section 5.5.4;

• the definition of $(\mathrm{\infty }\mathrm{,1})$-topoi of $(\mathrm{\infty }\mathrm{,1})$-sheaves is then definition 6.1.0.4 in section 6.1;

• the characterization of $(\mathrm{\infty }\mathrm{,1})$-sheaves in terms of descent and codescent is in section 6.1.3

• the relation between the Brown–Joyal–Jardine model and the general story is discussed at length in section 6.5.4