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

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos

Locality and descent



The term stack, is a traditional synonym for 2-sheaf or often, more restrictively, as a synonym for (2,1)-sheaf (see there for more details).

This is part of a whole hierarchy of higher categorical generalizations of the notion of sheaf. A (2,1)-sheaf / stack is equivalently a 1-truncated (∞,1)-sheaf/∞-stack.

Generally, then, n-stack is a synonym for (n+1)-sheaf, or more restrictively for (n+1,1)-sheaf.

More concretely this means that a 1-stack on a site SS (or more generally (2,1)-site or even (2,2)-site SS) is

If the pseudofunctor takes values in the 2-subcategory GrpdCat\subset Cat of groupoids, the stack is sometimes referred to as a stack of groupoids. This is the more commonly occurring case so the term ‘stack’ has come to mean ‘stack of groupoids’ in much of the literature.

In some circles the notion of a stack as a generalized groupoid is almost more familiar than the notion of sheaf as a generalized space. For instance differentiable stacks have attracted much attention in the study of Lie groupoids and orbifolds, while diffeological spaces are only beginning to be investigated more in Lie theory. Groupoid stacks are closely related to internal groupoids MO.

An algebraic stack, differentiable stack etc. is a stack over a site of schemes or differentiable manifolds with additional representability conditions.

Provisional discussion

The following is “provisional” material on stacks that Todd Trimble wrote in the course of a discussion with Urs. Somebody should turn this here into a coherent entry on stacks.

(Todd speaking.) I don’t really speak “stacks”, but in an effort to build a bridge between sheaves and stacks, I’ll write down what I thought I understood, and ask someone such as Urs to come in and check. (Warning: I’m treating this edit box almost as a sandbox, in that what I say below is all a bit provisional until we get some discussion going.)

Hi Todd, thanks for this. I started making some remarks on the relation between descent \infty-categories and pseudofunctors from covers regarded as sieves (hence as presheaves) at descent and codescent in the section titled Descent in terms of pseudo-functors.

At the simplest level, let CC be a category. As we know, a presheaf on CC is just a functor X:C opSetX: C^{op} \to Set.

Now let’s categorify just once: regard a category CC as a bicategory whose local hom-categories are discrete. What I’ll call a “pre-stack” is then a homomorphism of bicategories X:C opCatX: C^{op} \to Cat. Here I’m following Street’s terminology: a homomorphism of bicategories is the “pseudo” version of a weak map of bicategories, as opposed to the “lax” version. So, we have given coherent isomorphisms X(f)X(g)X(fg)X(f)X(g) \to X(f g), and so on.

Now suppose that CC also comes equipped with a topology JJ, and let FF be a JJ-covering sieve for cc, so that in particular it’s a subfunctor i:Fhom(,c)i: F \hookrightarrow \hom(-, c). We want to build a (truncated) simplicial object out of this, and to this end I’ll use some yoga which was basically developed in my Cafe post on the bar construction [perhaps this may go partway to addressing your most recent query there, Urs].

Namely, there is a canonical way of presenting FF as a colimit of representables. Officially, it’s given by a coend formula, but it’s probably more illuminating to think of it in terms of tensor products over CC:

hom C(,) CF()F()\hom_C(-, -) \otimes_C F(-) \cong F(-)

In the long-winded version, this says that FF is the coequalizer of a diagram having the form

c,dhom C(,c)×hom C(c,d)×F(d) chom C(,c)×F(c)F()\sum_{c, d} \hom_C(-, c) \times \hom_C(c, d) \times F(d) \stackrel{\to}{\to} \sum_c \hom_C(-, c) \times F(c) \to F(-)

where the more visible one of the two parallel arrows involves the contravariant action of CC on FF:

hom(c,d)×F(d)F(c)\hom(c, d) \times F(d) \to F(c)

and the less visible one uses CC acting on itself:

hom(,c)×hom(c,d)×F(d)hom(,d)×F(d)\hom(-, c) \times \hom(c, d) \times F(d) \to hom(-, d) \times F(d)

The point now is that this coequalizer diagram represents the tail end of a simplicial object (with F()F(-) appearing in dimension -1), which in the notation of the bar construction one could call B(C,C,F)B(C, C, F). Let me explain this last bit.

The point is that any category CC can be regarded as a monad in the bicategory of spans. The underlying span is of course

C 0domC 1codC 0C_0 \stackrel{dom}{\leftarrow} C_1 \stackrel{cod}{\to} C_0

and a presheaf FF on CC, as a discrete op-fibration, has an underlying span

C 0F1C_0 \leftarrow F \to 1

and is precisely an algebra over the monad CC. Then, given the data of a monad and an algebra over that monad, one proceeds to build the bar construction as a simplicial object, and I think this is probably the simplicial thingy we want to base the category of descent data on (given a pre-stack XX).

In fact, if memory serves the category of descent data can be efficiently expressed in bicategorical language as follows. The covering sieve FF becomes a homomorphism of bicategories by changing base from SetSet to CatCat:

C opFSetdiscreteCatC^{op} \stackrel{F}{\to} Set \stackrel{discrete}{\to} Cat

and, abbreviating discretediscrete to dd, it turns out that

Desc F(X)Nat(dF,X)Desc_F(X) \simeq Nat(d F, X)

where the thing on the right side is the category of strong (i.e., pseudo) natural transformations between the indicated bicategory homomorphisms.

In that case, the stack condition on XX becomes the statement that the canonical functor

X(c)YonedaNat(dhom(,c),X)Nat(dF,X)X(c) \stackrel{Yoneda}{\cong} Nat(d \hom(-, c), X) \to Nat(d F, X)

(where the first equivalence comes from the bicategorical Yoneda lemma, and the second functor is induced from the subfunctor i:Fhom(,c)i: F \to \hom(-, c)) is an equivalence for all JJ-covering sieves FF. This formulation connects up nicely, that is, is a straight categorification of what was put down in the entry sheaf.


  • The stack of BGBG is a functor Mfd opGpdMfd^{op} \to Gpd, sending UMfdU\in Mfd to the groupoid of GG-principal bundles over UU with GG-equivariant morphisms; sending UfVU\xrightarrow{f} V to the functor induced by pullbacks of principal bundles via ff. Then BGBG comes from the prestack BG p:Mfd opGpdBG^p: Mfd^{op} \to Gpd sending UMfdU\in Mfd to the groupoid of trivial principal bundles over UU with GG-equivariant morphisms (then it is just a GG-valued function UgGU\xrightarrow{g} G; sending UfVU\xrightarrow{f}V to the functor induced by pullbacks of principal bundles via ff. Then sheafification or stackification will give us BGBG back.

Special kinds of stacks include

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 concept originates, under the French term champ, in

and under the English term stack in

Further early discussion includes

  • Marta Bunge, Robert Pare, Stacks and equivalence of indexed categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no.4 (1979) (numdam)

  • Marta Bunge, Stack completions and Morita equivalence for categories in a topos, Cahiers de topologie et géométrie différentielle xx-4, (1979)

    401-436, (MR558106, numdam)

See also the references at 2-sheaf.


Discussion of stacks in their incarnation (under the Grothendieck construction) as Grothendieck fibrations.

  • Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory MR2223406; math.AG/0412512 pp. 1–104 in Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. 2005. x+339 pp. MR2007f:14001

A model category structure on presheaves of groupoids, presenting stacks, by a localization of a model structure on simplicial presheaves (modeling ∞-stacks before localization) is discussed in

Last revised on July 25, 2020 at 15:12:32. See the history of this page for a list of all contributions to it.