Contents

Idea

Recall that a locally constant sheaf (of sets) is a section of the constant stack with fiber the groupoid $\mathrm{Core}(\mathrm{FinSet})$, the core of the category FinSet.

This extends to a general pattern:

a locally constant $\mathrm{\infty }$-stack is a section of the constant ∞-stack that is constant on the ∞-groupoid $\mathrm{Core}(\mathrm{\infty }\mathrm{FinGrpd})$.

Definition

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

consisting of the global section and the constant ∞-stack (∞,1)-functor.

Write $𝒮:=\mathrm{core}(\mathrm{Fin}\mathrm{\infty }\mathrm{Grpd})\in \mathrm{\infty }\mathrm{Grpd}$ for the core ∞-groupoid of the (∞,1)-category of finite $\mathrm{\infty }$-groupoids. (We can drop the finiteness condition by making use of a larger universe.) This is canonically a pointed object $*\to 𝒮$.

Notice the for $X\in H$ any object, the over-(∞,1)-topos $H/X$ is the little $(\mathrm{\infty },1)$-topos of $X$. Objects in here we may regard as $\mathrm{\infty }$-stacks on $X$.

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

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 $\mathrm{LConst}$ 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 ${\mathrm{Sh}}_{(2,1)}(X)(X,\mathrm{LConst}(C))$, so that the theorem then gives the adjunction equivalence $\cdots \simeq \mathrm{Grpd}({\Pi }_1(X),C)$).

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. (Subjct verbatim to a remark as above.)

Locally constant $\mathrm{\infty }$-stacks on topological spaces

A discussion of locally constant $\mathrm{\infty }$-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)

of locally constant ∞-stacks on $X$ and Kan fibrations over the fundamental ∞-groupoid $\Pi (X)=\mathrm{Sing}(X)$.

But by the right Quillen functor $\mathrm{Id}:{\mathrm{sSet}}_{\mathrm{Quillen}}\to {\mathrm{sSet}}_{\mathrm{Joyal}}$ 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 $\Pi (X)\to \mathrm{\infty }\mathrm{Grpd}$.

In total this means that via the Grothendieck construction Toën’s result does actually produce an equivalence

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 $\mathrm{\infty }$-stack is a section of a constant ∞-stack.

A locally constant sheaf / $\mathrm{\infty }$-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.