(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 $\infty$-stack functor $LConst$ 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 $X$ as an ∞-stack on Top:
there is the ∞-stack $\bar const_X$ constant on $X$, meaning constant on the Kan complex that is the fundamental ∞-groupoid $Sing X = \Pi(X)$ of $X$;
there is the Yoneda embedding $Y(X)$ of $X$ into ∞-stack.
The first regards $X$ really as an ∞-groupoid, forgetting its topology, the second regards $X$ as a locale, not caring about the homotopies that are inside.
For any (∞,1)-category $S$, there is the obvious embedding of ∞-groupoids into (∞,1)-presheaves on $S$
where of course
for all $U$.
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 $S = Top$ itself, there are then two different ways to regard a topological space as an $\infty$-stack, and they have very different meaning.
In particular, with $X$ a topological space, the $\infty$-stack constant on $X$ has the property that its loop space object $\Lambda X$ is indeed the $\infty$-stack constant on the free loop space of $X$, while the loop space object of $X$ regarded as a representable $\infty$-stack is just $X$ itself again.
This is because
the $\infty$-stack represented by $X$ regards $X$ as a categorically discrete topological groupoid;
while the $\infty$-stack constant on $X$ regards $X$ 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 $\infty$-stack.
A locally constant sheaf / $\infty$-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.