(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The notion of, equivalently
and specifically of
is the $\infty$-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 $\infty$-stack if it “satisfies descent” in that its assignment to a space $X$ is equivalent to its descent data for any cover or hypercover $Y^\bullet \to X$: if the canonical morphism
is an equivalence. This is the descent condition.
One important motivation for $\infty$-stacks is that they generalize the notion of Grothendieck topos from 1-categorical to higher categorical context.
This is a central motivation for considering higher stacks. They may also be thought of as internal ∞-groupoids in a sheaf topos.
A well developed theory exists for $\infty$-stacks that are sheaves with values in ∞-groupoids. Given that ordinary sheaves may be thought of as sheaves of 0-categories and that $\infty$-groupoid-values sheaves may be thought of as sheaves of (∞,0)-categories, these may be called (∞,1)-sheaves. In the case that these $\infty$-groupoids have vanishing homotopy groups above some degree $n$, 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
of the (∞,1)-category of (∞,1)-presheaves with values in ∞Grpd, such that the left adjoint (∞,1)-functor $\bar {(\cdot)}$ – the ∞-stackification operation – is left exact.
One of the main theorems of Higher Topos Theory says that the old model structures on simplicial presheaves are the canonical
This allows to regard various old technical results in a new conceptual light and provides powerful tools for actually handling $\infty$-stacks.
In particular this implies that the old definition of abelian sheaf cohomology is secretly the computation of ∞-stackification for $\infty$-stacks that are in the image of the Dold-Kan embedding of chain complexes of sheaves into simplicial sheaves.
Notice that an $\infty$-stack is a (∞,1)-presheaf for which not only the codomain is an (∞,1)-category, but where also the domain, the site, may be an (∞,1)-category.
To emphasize that one considers $\infty$-stacks on higher categorical sites one speaks of derived stacks.
The above concerns $\infty$-stacks with values in ∞-groupoids, i.e, (∞,0)-categories. More generally there should be notions of $\infty$-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 $\infty$-stacks are quasicoherent ∞-stacks of modules, being the categorification of the notion of quasicoherent sheaf. By their nature these are really $(\infty,1)$-stacks in that they take values not in ∞-groupoids but in (∞,1)-categories, but often only their ∞-groupoidal core is considered.
In
for the site $C = Alg_k^{op}$ with a suitable topology a Quillen adjunction
is presented, where $\mathcal{O}$ sends and $\infty$-stack to its global dg-algebra of functions and $Spec$ constructs the simplicial presheaf “represented” degreewise by a simplicial algebra (under the monoidal Dold-Kan correspondence these are equivalent to dg-algebras).
An $\infty$-stack in the image of $Spec : dgAlg_k^+ \to sPSh(C)$ is an affine $\infty$-stack. The image of an arbitrary $\infty$-stack under the composite
is its affinization.
This notion was considered in the full (∞,1)-category picture in
where it is also generalized to derived stacks, i.e. to the (∞,1)-site $dgAlg_k^-$ of cochain dg-algebras in non-positive degree, where the pair of adjoint (∞,1)-functors is
with $\mathcal{O}$ taking values in unbounded dg-algebras.
In detail, $\mathcal{O}$ acts as follows: every ∞-stack $X$ may be written as a (colimit) over representable $Spec A_i \in dgAlg_i$
where $Y : (dgAlg^-)^{op} \to \mathbf{H}$ is the (∞,1)-Yoneda embedding.
The functor $\mathcal{O}$ takes any such colimit-description, and simply reinterprets the colimit in $dgAlg^{op}$, i.e. the limit in $dgAlg$:
(∞,1)-sheaf / $\infty$-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 |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
The study of $\infty$-stacks is known in parts as the study of nonabelian cohomology. See there for further references.
The search for $\infty$-stacks probably began with Alexander Grothendieck in Pursuing Stacks.
The notion of $\infty$-stacks can be set up in various notions of $\infty$-categories. Andre Joyal, Jardine, Bertrand Toen and others have developed the theory of $\infty$-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
Bertrand Toën, Gabriele Vezzosi; Segal topoi and stacks over Segal categories, math.AG/0212330.
Bertrand Toën; Higher and derived stacks: a global overview (arXiv).
This concerns $\infty$-stacks with values in ∞-groupoids, i.e. $(\infty,0)$-categories. More generally descent conditions for $n$-stacks and $(\infty,n)$-stacks with values in (∞,n)-categories have been earlier discussed in
All this has been embedded into a coherent global theory in the setting of quasicategories in