nLab geometric stack

Idea

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

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

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

Special cases

Geometric stacks for the following choices of sites $C$ are called

• for $C =$ Top – topological stack;

• for $C =$ Diff – differentiable stack;

• for $C =$ CRing${}^{op}$ – algebraic stack;

• for $C =$ CplxMfd – complex analytic stack;

• for $C =$ Loc – localic stack;

Definition

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

A general requirement is that

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

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

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

This is necessarily itself representable, precisely if $\Delta_X$ is.

Further conditions are the following

• for $C = Sch_{et}$ the site of schemes with the etale topology

  • $\Delta_X$ is required to be quasicompact and separated?

  • for Deligne-Mumford stacks $p$ is moreover required to be etale

  • for Artin stacks $p$ is required to be smooth.

Relation to groupoid objects

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

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

Related concepts

• geometric stack

• geometric ∞-stack

References

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

• Jochen Heinloth, Notes on differentiable stacks (2004) [pdf, pdf]

Differentiable stacks are discussed in

• Kai Behrend, Ping Xu, Differentiable Stacks and Gerbes (arXiv).

Specifically for the relation to groupoid objects see

3.1 and 3.3 in

• Metzler, Topological and smooth stacks (arXiv:math/0306176)

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

• Eugene Lerman, Orbifolds as stacks? (arXiv:0806.4160)

See also

• Bertrand Toen, Master course on algebraic stacks.

• Aise Johan de Jong, The Stacks Project (pdf) (project website)

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

• Dominic Joyce, Algebraic geometry over $C^\infty$-rings (arXiv:1001.0023)