higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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
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.
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;
There are slight variations in the literature on what precisely 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
the diagonal morphism $\Delta : X \to X \times X$ is a representable morphism of stacks
there exists an atlas for the stack, in that there is a representable $U \in C$ and a surjective morphism
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.
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
geometric stack
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
paragraph 4.4 in
See also
Bertrand Toen, Master Course on Stacks (web)
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