infinity-stack in nLab

Contents

Contents

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 XX is equivalent to its descent data for any cover or hypercover Y •→XY^\bullet \to X: if the canonical morphism

A(X)→Desc(Y •,A) \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 nn, 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

∞Stacks(C)≃Sh ∞(C)→←(⋅)¯PSh ∞(C) \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 (∞,1)(\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 opC = Alg_k^{op} with a suitable topology a Quillen adjunction

𝒪:sPSh(C) loc→←[Δ op,Alg k]≃dgAlg k +:Spec \mathcal{O} : sPSh(C)_{loc} \stackrel{\leftarrow}{\to} [\Delta^{op},Alg_k] \simeq dgAlg_k^{+} : Spec

is presented, where 𝒪\mathcal{O} sends and ∞\infty-stack to its global dg-algebra of functions and SpecSpec 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 +→sPSh(C)Spec : dgAlg_k^+ \to sPSh(C) is an affine ∞\infty-stack. The image of an arbitrary ∞\infty-stack under the composite

Aff:sPSh(C)→𝒪dgAlg k +→SpecsPSh(C) 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 −dgAlg_k^- of cochain dg-algebras in non-positive degree, where the pair of adjoint (∞,1)-functors is

𝒪:Sh (∞,1)((dgAlg k −) op)→←[Δ op,dgAlg k −]≃dgAlg k:Spec \mathcal{O} : Sh_{(\infty,1)}((dgAlg_k^-)^{op}) \stackrel{\leftarrow}{\to} [\Delta^{op},dgAlg_k^-] \simeq dgAlg_k : Spec

with 𝒪\mathcal{O} taking values in unbounded dg-algebras.

In detail, 𝒪\mathcal{O} acts as follows: every ∞-stack XX may be written as a (colimit) over representable SpecA i∈dgAlg iSpec A_i \in dgAlg_i

X≃lim → iY(SpecA i), X \simeq \lim_{\to^i} Y(Spec A_i) \,,

where Y:(dgAlg −) op→HY : (dgAlg^-)^{op} \to \mathbf{H} is the (∞,1)-Yoneda embedding.

The functor 𝒪\mathcal{O} takes any such colimit-description, and simply reinterprets the colimit in dgAlg opdgAlg^{op}, i.e. the limit in dgAlgdgAlg:

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

• sheaf

• 2-sheaf / stack

• (∞,1)-sheaf / ∞\infty-stack,

  • sheaf of spectra

  • sheaf of L-∞ algebras

• (∞,2)-sheaf

• (∞,n)-sheaf

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. (∞,0)(\infty,0)-categories. More generally descent conditions for nn-stacks and (∞,n)(\infty,n)-stacks with values in (∞,n)-categories have been earlier discussed in

• Andre Hirschowitz, Carlos Simpson; Descente pour les nn-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]