(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A constant ∞-stack/(∞,1)-sheaf is the ∞-stackification of a (∞,1)-presheaf which is constant as an (∞,1)-functor.
With the global section (∞,1)-functor the constant -stack functor forms the terminal (∞,1)-geometric morphism
Notice that in the special case of ∞-stacks on Top, hence of topological ∞-groupoid, which may be thought of as Top-valued presheaves on Top(!), there are two different obvious ways to regard a topological space as an ∞-stack on Top:
there is the ∞-stack constant on , meaning constant on the Kan complex that is the fundamental ∞-groupoid of ;
there is the Yoneda embedding of into ∞-stack.
The first regards really as an ∞-groupoid, forgetting its topology, the second regards as a locale, not caring about the homotopies that are inside.
For any (∞,1)-category , there is the obvious embedding of ∞-groupoids into (∞,1)-presheaves on
where of course
for all .
This is all very obvious, but deserves maybe a special remark in the case that ∞-groupoids are modeled as (compactly generated and weakly Hausdorff) topological spaces: in particular in the case that itself, there are then two different ways to regard a topological space as an -stack, and they have very different meaning.
In particular, with a topological space, the -stack constant on has the property that its loop space object is indeed the -stack constant on the free loop space of , while the loop space object of regarded as a representable -stack is just itself again.
This is because
the -stack represented by regards as a categorically discrete topological groupoid;
while the -stack constant on regards as a topologically discrete groupoid which however may have nontrivial morphisms.
A locally constant function is a section of a constant sheaf;
a locally constant sheaf is a section of a constant stack;
a locally constant stack is a section of (… and so on…)
a locally constant ∞-stack is a section of a constant -stack.
A locally constant sheaf / -stack is also called a local system.
Last revised on November 8, 2010 at 19:00:10. See the history of this page for a list of all contributions to it.