A stack on a site is geometric if, roughly, it is “represented” by a suitably well-behaved groupoid internal to , i.e. if to an object the stack assigns assigns the (ordinary) groupoid
Notice that a crucial difference between the groupoid object in and the geometric stack is that the equivalence class of the stack in general contains more (geometric) stacks than there are groupoid objects internally equivalent to : two groupoid objects with equivalent geometric stacks are called Morita equivalent groupoid objects.
With fixed, instead of “geometric stack” one says, for instance
for Top – topological stack;
for Diff – differentiable stack;
for CRing – algebraic stack;
There are slight variations in the literature on what particular is required of a stack on a site with subcabonical topology in order that it qualifies as geometric.
A general requirement is that
the diagonal morphism is a representable morphism of stacks
there exists an atlas for the stack, in that there is a representable and a surjective morphism
This is necessarily itself representable, precisely if is.
Further conditions are the following
for the site of schemes with the etale topology?
is required to be quasicompact and separated
for Deligne-Mumford stacks is moreover required to be etale
The groupoid object associated to a geometric stack with atlas is simply the 2-categorical Cech-nerve of : set and , where the latter is the 2-categorical pullback
A good discussion of topological and differentiable stacks is around definition 2.3 in
Specifically for the relation to groupoid objects see
paragraohs 2.4.3, 3.4.3, 3.8, 4.3 in
paragraph 4.4 in
See also
Geometric stacks over the site of schemes modeled on smooth loci is in section 8 of
Geometric -stacks were introduced in
in terms of model category presentations.
More discussion on that is in