(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 sheaf is a sheaf (on some site C) that is isomorphic to the sheafification of a constant presheaf (a constant functor).
Together with the global section functor, the constant sheaf functor is a geometric morphism
from the sheaf topos to the topos Set.
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 / $\infty$-stack is also called a local system.
See also at constant presheaf.