# nLab constant infinity-stack

### Context

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

cohomology

# Contents

## Definition

A constant ∞-stack/(∞,1)-sheaf is the ∞-stackification of a (∞,1)-presheaf which is constant as an (∞,1)-functor.

With the global section (∞,1)-functor the constant $\infty$-stack functor $\mathrm{LConst}$ forms the terminal (∞,1)-geometric morphism

$\left(\mathrm{LConst}⊣\Gamma \right):{\mathrm{Sh}}_{\left(\infty ,1\right)}\left(C\right)\stackrel{\stackrel{\mathrm{LConst}}{←}}{\underset{\Gamma }{\to }}\infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$(LConst \dashv \Gamma) : Sh_{(\infty,1)}(C) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,.

## On $\mathrm{Top}$

Notice that in the special case of ∞-stacks on Top, hence of topological ∞-groupoid, which may be thought of as Top-valued presheaves on Top(!), there are two different obvious ways to regard a topological space $X$ as an ∞-stack on Top:

• there is the ∞-stack ${\overline{\mathrm{const}}}_{X}$ constant on $X$, meaning constant on the Kan complex that is the fundamental ∞-groupoid $\mathrm{Sing}X=\Pi \left(X\right)$ of $X$;

• there is the Yoneda embedding $Y\left(X\right)$ of $X$ into ∞-stack.

The first regards $X$ really as an ∞-groupoid, forgetting its topology, the second regards $X$ as a locale, not caring about the homotopies that are inside.

For any (∞,1)-category $S$, there is the obvious embedding of ∞-groupoids into (∞,1)-presheaves on $S$

$\mathrm{const}:\infty \mathrm{Grpd}\to \left[{S}^{\mathrm{op}},\infty \mathrm{Grpd}\right]$const : \infty Grpd \to [S^{op}, \infty Grpd]

where of course

${\mathrm{const}}_{K}:U↦K$const_K : U \mapsto K

for all $U$.

This is all very obvious, but deserves maybe a special remark in the case that ∞-groupoids are modeled as (compactly generated and weakly Hausdorff) topological spaces: in particular in the case that $S=\mathrm{Top}$ itself, there are then two different ways to regard a topological space as an $\infty$-stack, and they have very different meaning.

In particular, with $X$ a topological space, the $\infty$-stack constant on $X$ has the property that its loop space object $\Lambda X$ is indeed the $\infty$-stack constant on the free loop space of $X$, while the loop space object of $X$ regarded as a representable $\infty$-stack is just $X$ itself again.

This is because

• the $\infty$-stack represented by $X$ regards $X$ as a categorically discrete topological groupoid;

• while the $\infty$-stack constant on $X$ regards $X$ as a topologically discrete groupoid which however may have nontrivial morphisms.

## Pattern

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

Revised on November 8, 2010 19:00:10 by Urs Schreiber (131.211.232.76)