(,1)(\infty,1)-Topos theory

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos

Locality and descent



The notion of, equivalently

  • \infty-stack,

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 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.


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

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

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


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 simplicial 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

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 idgAlg iSpec A_i \in dgAlg_i

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

where Y:(dgAlg ) opHY : (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 \,.
homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\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.

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

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

Last revised on January 19, 2017 at 10:09:17. See the history of this page for a list of all contributions to it.