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locally constant infinity-stack

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(,1)(\infty,1)-Topos Theory

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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

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

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

Write 𝒮:=core(FinGrpd)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 XHX \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.

Definition

For XHX \in \mathbf{H} an object a locally constant \infty-stack on XX is an morphism XLConst𝒮X \to LConst \mathcal{S}.

The ∞-groupoid of locally constant \infty-stacks on XX is

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

An an object of the little (∞,1)-topos of XX, the over-(∞,1)-topos H/X\mathbf{H}/X the locally constant \infty-stack given by ˜\tilde \nabla is its (∞,1)-Grothendieck construction in H\mathbf{H}

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

the pullback of the universal fibration of finite ∞-groupoids

P LConst𝒵 X ˜ LConst𝒮. \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

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

(LConstΓ):Sh (2,1)(X)ΓLConstGrpd (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 1LConst):Op(X)Sh (2,1)(X)Π 1LConstGrpd (\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 2LConst):Op(X)Sh (3,1)(X)Π 2LConstGrpd (\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. (Subjct 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

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 QuillensSet 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 sheaf / \infty-stack is also called a local system.

References

Section A.1 of

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

Revised on April 4, 2014 23:17:34 by Tim Porter (2.26.36.99)