nLab geometric stack




A stack XX on a site CC is geometric if, roughly, it is represented by a suitably well-behaved groupoid object 𝒢=(𝒢 1𝒢 0)\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0) internal to CC, i.e. if to an object UCU \in C the stack assigns the (ordinary) groupoid

X:U(C(U,𝒢 1)C(U,𝒢 0)). X : U \mapsto (C(U,\mathcal{G}_1) \stackrel{\longrightarrow}{\longrightarrow} C(U,\mathcal{G}_0)) \,.

A crucial difference between the groupoid object 𝒢\mathcal{G} in CC and the geometric stack XX is that the equivalence class of the stack in general contains more (geometric) stacks than there are groupoid objects internally equivalent to 𝒢\mathcal{G}: two groupoid objects with equivalent geometric stacks are called Morita equivalent groupoid objects.

Special cases

Geometric stacks for the following choices of sites CC are called


There are slight variations in the literature on what precisely is required of a stack XX on a site CC with subcanonical topology in order that it qualifies as geometric.

A general requirement is that

  1. the diagonal morphism Δ:XX×X\Delta : X \to X \times X is a representable morphism of stacks

  2. there exists an atlas for the stack, in that there is a representable UCU \in C and a surjective morphism

p:UX. p : U \to X \,.

This is necessarily itself representable, precisely if Δ X\Delta_X is.

Further conditions are the following

Relation to groupoid objects

The groupoid object associated to a geometric stack XX with atlas p:UXp : U \to X is the Cech groupoid of pp (this is simply the Cech groupoid of pp seen as a singleton cover) defined by 𝒢 0:=U\mathcal{G}_0 := U and 𝒢 1=U× XU\mathcal{G}_1 = U \times_X U, where the latter is the 2-categorical pullback

𝒢 1 s U t p U X \array{ \mathcal{G}_1 &\stackrel{s}{\to}& U \\ \downarrow^{\mathrlap{t}} &{}^{\simeq}\swArrow& \downarrow^{\mathrlap{p}} \\ U &\to& X }


A good discussion of topological and differentiable stacks is around definition 2.3 in

Differentiable stacks are discussed in

Specifically for the relation to groupoid objects see

3.1 and 3.3 in

paragraphs 2.4.3, 3.4.3, 3.8, 4.3 in

  • G. Laumon, L. Moret-Bailly, Champs algébriques , Ergebn. der Mathematik und ihrer Grenzgebiete 39 , Springer-Verlag, Berlin, 2000

paragraph 4.4 in

See also

Geometric stacks over the site of schemes modeled on smooth loci is in section 8 of

Last revised on April 15, 2023 at 19:33:00. See the history of this page for a list of all contributions to it.