nLab infinity-stack

Idea

The notion of, equivalently

• $\infty$-stack,

and specifically of

• (∞,1)-sheaf,

• geometric homotopy type

is the $\infty$-categorification of the notion of, equivalently

• sheaf and stack

• geometric homotopy type.

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

$$\mathbf{A}(X) \to Desc(Y^\bullet, \mathbf{A})$$

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.

Definition

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

• Jacob Lurie, Higher Topos Theory

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

$$\infty Stacks(C) \simeq Sh_\infty(C) \stackrel{\stackrel{\bar{(\cdot)}}{\leftarrow}}{\to} PSh_\infty(C)$$

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

• models for ∞-stack (∞,1)-toposes.

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.

Derived $\infty$-stacks

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.

Higher $\infty$-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.

Quasicoherent $\infty$-stacks

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.

Affine $\infty$-stacks

In

• Bertrand Toen, Affine stacks (Champs affines) (arXiv:math/0012219)

for the site $C = Alg_k^{op}$ with a suitable topology a Quillen adjunction

$$\mathcal{O} : sPSh(C)_{loc} \stackrel{\leftarrow}{\to} [\Delta^{op},Alg_k] \simeq dgAlg_k^{+} : Spec$$

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 cosimplicial 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

$$Aff : sPSh(C) \stackrel{\mathcal{O}}{\to} dgAlg_k^+ \stackrel{Spec}{\to} sPSh(C)$$

is its affinization.

This notion was considered in the full (∞,1)-category picture in

• David Ben-Zvi, David Nadler, Loop Spaces and Connections (arXiv:math/1002.3636)

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

$$\mathcal{O} : Sh_{(\infty,1)}((dgAlg_k^-)^{op}) \stackrel{\leftarrow}{\to} [\Delta^{op},dgAlg_k^-] \simeq dgAlg_k : Spec$$

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 $Spec A_i \in dgAlg_i$

$$X \simeq \lim_{\to^i} Y(Spec A_i) \,,$$

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 $dgAlg$:

$$\mathcal{O}(X) = \lim_{\leftarrow^i} A_i \,.$$

Related concepts

• sheaf

• 2-sheaf / stack

• (∞,1)-sheaf / $\infty$-stack,

  • sheaf of spectra

  • sheaf of L-∞ algebras

• (∞,2)-sheaf

• (∞,n)-sheaf

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

References

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, In: Algebraic Geometry Seattle 2005, Proceedings of Symposia in Pure Mathematics, Vol. 80.1, AMS 2009 (arXiv:math/0604504, doi:10.1090/pspum/080.1)

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

• Andre Hirschowitz, Carlos Simpson; Descente pour les $n$-champs (arXiv)

All this has been embedded into a coherent global theory in the setting of quasicategories in

• Jacob Lurie, Higher Topos Theory

Textbook account on presentation by model structures on simplicial presheaves:

• John F. Jardine, Local homotopy theory, Springer Monographs in Mathematics (2015) [doi:10.1007/978-1-4939-2300-7]