nLab locally constant infinity-stack

Idea

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 $Core(\infty FinGrpd)$.

Definition

For $\mathbf{H}$ an (∞,1)-sheaf (∞,1)-topos there is the terminal (∞,1)-geometric morphism

$$(LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd$$

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 $X$.

Definition

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

$$LConst(X) := \mathbf{H}(X, LConst \mathcal{S}) \,.$$

Remark

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 $\mathbf{H}$

$$\array{ P &\to& * \\ \downarrow &\swArrow& \downarrow \\ X &\stackrel{\tilde \nabla}{\to}& LConst \mathcal{S} }$$

the pullback of the universal fibration of finite ∞-groupoids

$$\array{ P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow \\ X &\stackrel{\tilde \nabla}{\to}& LConst \mathcal{S} } \,.$$

Remark

A locally constant $\infty$-stack is also called a local system. See there for more details.

Examples

Here are commented references that establish aspects of the above general abstract situation.

Locally constant 1-stacks and 2-stacks on topological spaces

A discussion of locally constant 2-stacks over topological spaces is in

• Pietro Polesello and Ingo Waschkies, Higher monodromy , Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150 (pdf).

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

$$(LConst \dashv \Gamma) : Sh_{(2,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} Grpd$$

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

$$(\Pi_1\dashv LConst) : Op(X) \hookrightarrow Sh_{(2,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Pi_1}{\to}} Grpd$$

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

$$(\Pi_2\dashv LConst) : Op(X) \hookrightarrow Sh_{(3,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Pi_2}{\to}} Grpd$$

now with the path 2-groupoid operation (locally) left adjoint to forming constant 2-stacks. (Subject verbatim to a remark as above.)

Locally constant $\infty$-stacks on topological spaces

A discussion of locally constant $\infty$-stacks over topological spaces is in

• Bertrand Toen, Toward a Galoisian interpretation of homotopy theory (arXiv:0007157)

In theorem 2.13, p. 25 the author proves an equivalence of (∞,1)-categories (modeled there as Segal categories)

$$LConst(X) \simeq Fib(\Pi(X))$$

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

$$LConst(X) \simeq Func(\Pi(X), \infty Grpd) \,.$$

Pattern

• 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.

References

Section A.1 of

• Jacob Lurie, Higher Algebra .

See also the references at geometric homotopy groups in an (∞,1)-topos.