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Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## 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

## 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

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

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