derived smooth geometry
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.
Geometric stacks for the following choices of sites are called
A general requirement is that
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
The groupoid object associated to a geometric stack with atlas is the Cech groupoid? of (this is simply the Cech groupoid of seen as a singleton cover) defined by and , where the latter is the 2-categorical pullback
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
Geometric stacks over the site of schemes modeled on smooth loci is in section 8 of