structures in a cohesive (∞,1)-topos
The notion of, equivalently
and specifically of
is the -categorification of the notion of, equivalently
Where a sheaf is a presheaf with values in Set that satisfies the sheaf condition, an ∞-category-valued (pseudo)presheaf is an -stack if it “satisfies descent” in that its assignment to a space is equivalent to its descent data for any cover or hypercover : if the canonical morphism
is an equivalence. This is the descent condition.
A well developed theory exists for -stacks that are sheaves with values in ∞-groupoids. Given that ordinary sheaves may be thought of as sheaves of 0-categories and that -groupoid-values sheaves may be thought of as sheaves of (∞,0)-categories, these may be called (∞,1)-sheaves. In the case that these -groupoids have vanishing homotopy groups above some degree , these are sometimes also called sheaf of n-types.
The currently most complete picture of (∞,1)-sheaves appears in
but is based on a long development by other authors, some of which is indicated in the list of references below.
With the general machinery of (∞,1)-category theory in place, the definition of the (∞,1)-category of ∞-stacks is literally the same as that of a category of sheaves: it is a reflective (∞,1)-subcategory
This allows to regard various old technical results in a new conceptual light and provides powerful tools for actually handling -stacks.
In particular this implies that the old definition of abelian sheaf cohomology is secretly the computation of ∞-stackification for -stacks that are in the image of the Dold-Kan embedding of chain complexes of sheaves into simplicial sheaves.
To emphasize that one considers -stacks on higher categorical sites one speaks of derived stacks.
The above concerns -stacks with values in ∞-groupoids, i.e, (∞,0)-categories. More generally there should be notions of -stacks with values in (n,r)-categories. These are expected to be modeled by the model structure on homotopical presheaves with values in the category of Theta spaces.
An archetypical class of examples of -stacks are quasicoherent ∞-stacks of modules, being the categorification of the notion of quasicoherent sheaf. By their nature these are really -stacks in that they take values not in ∞-groupoids but in (∞,1)-categories, but often only their ∞-groupoidal core is considered.
is presented, where sends and -stack to its global dg-algebra of functions and constructs the simplicial presheaf “represented” degreewise by a simplicial algebra (under the monoidal Dold-Kan correspondence these are equivalent to dg-algebras).
An -stack in the image of is an affine -stack. The image of an arbitrary -stack under the composite
is its affinization.
This notion was considered in the full (∞,1)-category picture in
with taking values in unbounded dg-algebras.
where is the (∞,1)-Yoneda embedding.
The functor takes any such colimit-description, and simply reinterprets the colimit in , i.e. the limit in :
(∞,1)-sheaf / -stack,
|homotopy level||n-truncation||homotopy theory||higher category theory||higher topos theory||homotopy type theory|
|h-level 0||(-2)-truncated||contractible space||(-2)-groupoid||true/unit type/contractible type|
|h-level 1||(-1)-truncated||contractible-if-inhabited||(-1)-groupoid/truth value||(0,1)-sheaf/ideal||mere proposition/h-proposition|
|h-level 2||0-truncated||homotopy 0-type||0-groupoid/set||sheaf||h-set|
|h-level 3||1-truncated||homotopy 1-type||1-groupoid/groupoid||(2,1)-sheaf/stack||h-groupoid|
|h-level 4||2-truncated||homotopy 2-type||2-groupoid||(3,1)-sheaf/2-stack||h-2-groupoid|
|h-level 5||3-truncated||homotopy 3-type||3-groupoid||(4,1)-sheaf/3-stack||h-3-groupoid|
The study of -stacks is known in parts as the study of nonabelian cohomology. See there for further references.
The notion of -stacks can be set up in various notions of -categories. Andre Joyal, Jardine, Bertrand Toen and others have developed the theory of -stacks in the context of simplicial presheaves and also in Segal categories.
Bertrand Toën, Gabriele Vezzosi; Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193 (2005), no. 2, 257–372, doi, Homotopical Algebraic Geometry II: geometric stacks and applications, math.AG/0404373
All this has been embedded into a coherent global theory in the setting of quasicategories in