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locally constant infinity-stack

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

We propose a definition of locally constant $\mathrm{\infty }$-stacks inside any (∞,1)-topos.

Urs Schreiber: please somebody sanity-check the following.

Let $C$ be some site and let $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ be the (∞,1)-topos of ∞-stacks over $C$. We think of objects $X\in H$ as generalized spaces modeled on $C$.

We have a squence of adjunctions

$${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\stackrel{\stackrel{L}{\leftarrow }}{\hookrightarrow }{\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\stackrel{\stackrel{\mathrm{const}}{\leftarrow }}{\to }{\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(*)=\mathrm{\infty }\mathrm{Grpd}:\mathrm{LConst}$$Sh_{(\infty,1)}(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \stackrel{\overset{const}{\leftarrow}}{\to} PSh_{(\infty,1)}(*) = \infty Grpd : LConst

where on the left we have the defining adjunction of the (∞,1)-category of (∞,1)-sheaves – $L$ is ∞-stackification – and on the right the geometric morphism that is induced from the canonical morphism of sites $C\to *$ (to the terminal site).

Let

$$\mathrm{\infty }\mathrm{FinBund}:=\mathrm{LConst}(\mathrm{Core}(\mathrm{Fin}\mathrm{\infty }\mathrm{Grpd}))\in H$$\infty FinBund := LConst(Core(Fin \infty Grpd)) \in \mathbf{H}

be the constant ∞-stack that is constant on the ∞-groupoid which is the core of the (∞,1)-category of finite ∞-groupoids. We think of this as the $\mathrm{\infty }$-stack of $\mathrm{\infty }$-bundles with finite fibers – the generalization of the notion of covering space to the context $H$.

This is naturally a pointed object with point

$$(*\to \mathrm{\infty }\mathrm{FinBund}):=\mathrm{LConst}(*\to \mathrm{Core}(\mathrm{\infty }\mathrm{FinGrpd}))$$({*} \to \infty FinBund) \;:=\; LConst({*} \to Core(\infty FinGrpd))

the ∞-stackification of the inclusion of constant ∞-presheaves induced from the canonical inclusion $*\hookrightarrow$ ∞Grpd, identifying the terminal ∞-groupoid.

For $X\in H$ any object, the following terminology is suggestive:

• the $\mathrm{\infty }\mathrm{FinBund}$-principal ∞-bundle classified by a cocycle $X\to \mathrm{\infty }\mathrm{FinBund}$ is a covering $\mathrm{\infty }$-bundle of $X$;

• the ∞-groupoid

  $$\mathrm{\infty }\mathrm{FinBund}(X):=H(X,\mathrm{\infty }\mathrm{FinBund})$$\infty FinBund(X) := \mathbf{H}(X,\infty FinBund)

  is the $\mathrm{\infty }$-groupoid of covering $\mathrm{\infty }$-bundles on $X$;

• hence the cohomology $H(X,\mathrm{\infty }\mathrm{FinBund})={\pi }_{0}H(X,\mathrm{\infty }\mathrm{FinBund})$ is the set of equivalence classes of covering $\mathrm{\infty }$-bundles of $X$.

If the left adjoint $\mathrm{LConst}$ is also a right adjoint

$${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Pi }{\to }}\mathrm{\infty }\mathrm{Grpd}$$Sh_{(\infty,1)}(C) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Pi}{\to}} \infty Grpd

then we have

$$\mathrm{\infty }\mathrm{FinBund}(X)\simeq \mathrm{Func}(\Pi (X),\mathrm{\infty }\mathrm{FinGrpd}).$$\infty FinBund(X) \simeq Func(\Pi(X),\infty FinGrpd) \,.

This may be used to define the geometric homotopy group (of an ∞-stack).

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

$$(\mathrm{LConst}⊣\Gamma ):{\mathrm{Sh}}_{(\mathrm{2,1})}(X)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{Grpd}$$(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

$$({\Pi }_1⊣\mathrm{LConst}):\mathrm{Op}(X)\hookrightarrow {\mathrm{Sh}}_{(\mathrm{2,1})}(X)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{{\Pi }_1}{\to }}\mathrm{Grpd}$$(\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 $\mathrm{LConst}$ 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 ${\mathrm{Sh}}_{(\mathrm{2,1})}(X)(X,\mathrm{LConst}(C))$, 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

$$({\Pi }_2⊣\mathrm{LConst}):\mathrm{Op}(X)\hookrightarrow {\mathrm{Sh}}_{(\mathrm{3,1})}(X)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{{\Pi }_2}{\to }}\mathrm{Grpd}$$(\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 $\mathrm{\infty }$-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)

$$\mathrm{LConst}(X)\simeq \mathrm{Fib}(\Pi (X))$$LConst(X) \simeq Fib(\Pi(X))

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

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

Very similar statements are discussed in

• Mike Shulman, Parametrized spaces model locally constant homotopy sheaves (arXiv:0706.2874)

and, building on that, in example 1.8 of

• Denis-Charles Cisinski, Locally constant functors , Math. Proc. Camb. Phil. Soc. , 147 (pdf)