nLab
geometric stack

Idea

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

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

Notice that a crucial difference between the groupoid object $𝒢$ 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 $𝒢$: two groupoid objects with equivalent geometric stacks are called Morita equivalent groupoid objects.

With $C$ fixed, instead of “geometric stack” one says, for instance

• for $C=$ Top – topological stack;

• for $C=$ Diff – differentiable stack;

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

Definition

There are slight variations in the literature on what particular is required of a stack $X$ on a site $C$ with subcabonical 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

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

This is necessarily itself representable, precisely if ${\Delta }_X$ is.

Further conditions are the following

• for $C={\mathrm{Sch}}_{\mathrm{et}}$ 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 stack?s $p$ 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 simply the 2-categorical Cech-nerve of $p$: set ${𝒢}_0:=U$ and ${𝒢}_1=U{\times }_{X}U$, where the latter is the 2-categorical pullback

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

References

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

• Jochen Heinloth, Some notes on differentiable stacks (pdf)

Specifically for the relation to groupoid objects see

1. 1 and 3.3 in

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

paragraohs 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 Stacks (web)

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

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

Geometric $\mathrm{\infty }$-stacks

Geometric $\mathrm{\infty }$-stacks were introduced in

• Bertrand Toen, Affine stacks (Champs affines) (arXiv:math/0012219)

in terms of model category presentations.

More discussion on that is in

• Bertrand Toen, Gabriele Vezzosi, Homotopical Algebraic Geometry II: geometric stacks and applications (arXiv)