(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
Recall that a locally constant sheaf (of sets) is a section of the constant stack with fiber the groupoid $Core(FinSet)$, the core of the category FinSet.
This extends to a general pattern:
a locally constant $\infty$-stack is a section of the constant ∞-stack that is constant on the ∞-groupoid $Core(\infty FinGrpd)$.
For $\mathbf{H}$ an (∞,1)-sheaf (∞,1)-topos there is the terminal (∞,1)-geometric morphism
consisting of the global section and the constant ∞-stack (∞,1)-functor.
Write $\mathcal{S} := core(Fin \infty Grpd) \in \infty Grpd$ for the core ∞-groupoid of the (∞,1)-category of finite $\infty$-groupoids. (We can drop the finiteness condition by making use of a larger universe.) This is canonically a pointed object $* \to \mathcal{S}$.
Notice the for $X \in \mathbf{H}$ any object, the over-(∞,1)-topos $\mathbf{H}/X$ is the little $(\infty,1)$-topos of $X$. Objects in here we may regard as $\infty$-stacks on $X$.
For $X \in \mathbf{H}$ an object a locally constant $\infty$-stack on $X$ is an morphism $X \to LConst \mathcal{S}$.
The ∞-groupoid of locally constant $\infty$-stacks on $X$ is
An an object of the little (∞,1)-topos of $X$, the over-(∞,1)-topos $\mathbf{H}/X$ the locally constant $\infty$-stack given by $\tilde \nabla$ is its (∞,1)-Grothendieck construction in $\mathbf{H}$
the pullback of the universal fibration of finite ∞-groupoids
A locally constant $\infty$-stack is also called a local system. See there for more details.
Here are commented references that establish aspects of the above general abstract situation.
A discussion of locally constant 2-stacks over topological spaces is in
We indicate briefly how the results stated in this article fit into the general abstract picture as indicated above:
The authors consider locally constant 1-stacks and 2-stacks on sites of open subsets of topological spaces.
Prop. 1.1.9 gives the adjunction
between forming constant stacks and taking global sections.
Then prop 1.2.5, 1.2.6, culminating in theorem 1.2.9, p. 121 gives (somewhat implicitly) the other adjunction
with the right adjoint to $LConst$ being the fundamental groupoid functor on representables. (Where we change a bit the perspective on the results as presented there, to amplify the pattern indicated above. For instance where the authors write $\Gamma(X,C_X)$ we think of this here equivalently as $Sh_{(2,1)}(X)(X,LConst(C))$, so that the theorem then gives the adjunction equivalence $\cdots \simeq Grpd(\Pi_1(X),C)$).
Then in essentially verbatim analogy, these results are lifted from stacks to 2-stacks in section 2, where now prop 2.2.2, 2.2.3, culminating in theorem 2.2.5, p. 132 gives (somewhat implicitly) the adjunction
now with the path 2-groupoid operation (locally) left adjoint to forming constant 2-stacks. (Subject verbatim to a remark as above.)
A discussion of locally constant $\infty$-stacks over topological spaces is in
In theorem 2.13, p. 25 the author proves an equivalence of (∞,1)-categories (modeled there as Segal categories)
of locally constant ∞-stacks on $X$ and Kan fibrations over the fundamental ∞-groupoid $\Pi(X) = Sing(X)$.
But by the right Quillen functor $Id : sSet_{Quillen} \to sSet_{Joyal}$ from the Quillen model structure on simplicial sets to the Joyal model structure on simplicial sets every Kan fibration is a categorical fibration and every categorical fibration over a Kan complex is a Cartesian fibration (as discussed there) and a coCartesian fibration. Finally, by the (∞,1)-Grothendieck construction, these are equivalent to (∞,1)-functors $\Pi(X) \to \infty Grpd$.
In total this means that via the Grothendieck construction Toën’s result does actually produce an equivalence
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 $\infty$-stack is a section of a constant ∞-stack.
A locally constant sheaf / $\infty$-stack is also called a local system.
Section A.1 of
See also the references at geometric homotopy groups in an (∞,1)-topos.
Last revised on August 6, 2017 at 06:33:05. See the history of this page for a list of all contributions to it.