nLab
constant infinity-stack

Context

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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 LConstLConst forms the terminal (∞,1)-geometric morphism

(LConstΓ):Sh (,1)(C)ΓLConstGrpd. (LConst \dashv \Gamma) : Sh_{(\infty,1)}(C) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,.

On TopTop

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 XX as an ∞-stack on Top:

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

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

const:Grpd[S op,Grpd] const : \infty Grpd \to [S^{op}, \infty Grpd]

where of course

const K:UK const_K : U \mapsto K

for all UU.

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=TopS = 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 XX a topological space, the \infty-stack constant on XX has the property that its loop space object ΛX\Lambda X is indeed the \infty-stack constant on the free loop space of XX, while the loop space object of XX regarded as a representable \infty-stack is just XX itself again.

This is because

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

  • while the \infty-stack constant on XX regards XX 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)