# nLab locally constant infinity-stack

### Context

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

cohomology

# Contents

## Idea

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

## Definition

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

$\left(\mathrm{LConst}⊣\Gamma \right):H\to \infty \mathrm{Grpd}$(LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd

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

Write $𝒮:=\mathrm{core}\left(\mathrm{Fin}\infty \mathrm{Grpd}\right)\in \infty \mathrm{Grpd}$ 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 $*\to 𝒮$.

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

###### Definition

For $X\in H$ an object a locally constant $\infty$-stack on $X$ is an morphism $X\to \mathrm{LConst}𝒮$.

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

$\mathrm{LConst}\left(X\right):=H\left(X,\mathrm{LConst}𝒮\right)\phantom{\rule{thinmathspace}{0ex}}.$LConst(X) := \mathbf{H}(X, LConst \mathcal{S}) \,.
###### Remark

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

$\begin{array}{ccc}P& \to & *\\ ↓& ⇙& ↓\\ X& \stackrel{\stackrel{˜}{\nabla }}{\to }& \mathrm{LConst}𝒮\end{array}$\array{ P &\to& * \\ \downarrow &\swArrow& \downarrow \\ X &\stackrel{\tilde \nabla}{\to}& LConst \mathcal{S} }

the pullback of the universal fibration of finite ∞-groupoids

$\begin{array}{ccc}P& \to & \mathrm{LConst}𝒵\\ ↓& & ↓\\ X& \stackrel{\stackrel{˜}{\nabla }}{\to }& \mathrm{LConst}𝒮\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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.

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

$\left({\Pi }_{1}⊣\mathrm{LConst}\right):\mathrm{Op}\left(X\right)↪{\mathrm{Sh}}_{\left(2,1\right)}\left(X\right)\stackrel{\stackrel{\mathrm{LConst}}{←}}{\underset{{\Pi }_{1}}{\to }}\mathrm{Grpd}$(\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 $\Gamma \left(X,{C}_{X}\right)$ we think of this here equivalently as ${\mathrm{Sh}}_{\left(2,1\right)}\left(X\right)\left(X,\mathrm{LConst}\left(C\right)\right)$, so that the theorem then gives the adjunction equivalence $\cdots \simeq \mathrm{Grpd}\left({\Pi }_{1}\left(X\right),C\right)$).

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

$\left({\Pi }_{2}⊣\mathrm{LConst}\right):\mathrm{Op}\left(X\right)↪{\mathrm{Sh}}_{\left(3,1\right)}\left(X\right)\stackrel{\stackrel{\mathrm{LConst}}{←}}{\underset{{\Pi }_{2}}{\to }}\mathrm{Grpd}$(\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 $\infty$-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)

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

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

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 $\Pi \left(X\right)\to \infty \mathrm{Grpd}$.

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

$\mathrm{LConst}\left(X\right)\simeq \mathrm{Func}\left(\Pi \left(X\right),\infty \mathrm{Grpd}\right)\phantom{\rule{thinmathspace}{0ex}}.$LConst(X) \simeq Func(\Pi(X), \infty Grpd) \,.

## Pattern

A locally constant sheaf / $\infty$-stack is also called a local system.

Section A.1 of