locally constant infinity-stack in nLab

Contents

Contents

Idea

Recall that a locally constant sheaf (of sets) is a section of the constant stack with fiber the groupoid Core(FinSet)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(∞FinGrpd)Core(\infty FinGrpd).

Definition

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

(LConst⊣Γ):H→∞Grpd (LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd

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

Write 𝒮:=core(Fin∞Grpd)∈∞Grpd\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 *→𝒮* \to \mathcal{S}.

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

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⊣Γ):Sh (2,1)(X)→Γ←LConstGrpd (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

(Π 1⊣LConst):Op(X)↪Sh (2,1)(X)→Π 1←LConstGrpd (\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 LConstLConst 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 Γ(X,C X)\Gamma(X,C_X) we think of this here equivalently as Sh (2,1)(X)(X,LConst(C))Sh_{(2,1)}(X)(X,LConst(C)), so that the theorem then gives the adjunction equivalence ⋯≃Grpd(Π 1(X),C)\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

(Π 2⊣LConst):Op(X)↪Sh (3,1)(X)→Π 2←LConstGrpd (\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)≃Fib(Π(X)) LConst(X) \simeq Fib(\Pi(X))

of locally constant ∞-stacks on XX and Kan fibrations over the fundamental ∞-groupoid Π(X)=Sing(X)\Pi(X) = Sing(X).

But by the right Quillen functor Id:sSet Quillen→sSet JoyalId : 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 Π(X)→∞Grpd\Pi(X) \to \infty Grpd.

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

LConst(X)≃Func(Π(X),∞Grpd). 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.