nLab
locally contractible space

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Definition

A topological space XX is said to be locally contractible if it has a basis of open subsets that consists of contractible topological spaces UXU \hookrightarrow X.

Sometimes one requires just that the inclusions UXU \to X are null-homotopic map?s. This might be called semi-locally contractible.

Remark

One could also consider a basis of open sets such that the opens UU have (just) trivial homotopy groups, but this does not seem to crop up in practice.

Definition

A locale XX is locally contractible if, viewing a locale as a (0,1)(0,1)-topos and hence a (very special kind of) (,1)(\infty,1)-topos, it is locally ∞-connected.

Is this right? Do these two definitions correspond in that a sober space or topological locale is locally contractible as a topological space iff it's locally contractible as a locale? —Toby

Examples

Properties

Proposition

For XX a locally contractible topological space, the (∞,1)-category of (∞,1)-sheaves Sh (,1)(X)Sh_{(\infty,1)}(X) is a locally ∞-connected (∞,1)-topos.

This is discussed at locally ∞-connected (∞,1)-site.

Other viewpoints

If one considers fundamental ∞-groupoids, the inclusion UXU \to X being null-homotopic is equivalent to the induced (∞,1)-functor Π(U)Π(X)\Pi(U) \to \Pi(X) being naturally isomorphic to the trivial functor sending everything to a single point.

David Roberts: The following may be straightforwardly obvious, but I have couched it as a conjecture, because I haven’t seen it in print.

Conjecture

If the space XX is semi-locally contractible then every locally constant nn-stack on the site of open sets of XX is locally trivial.

See also locally ∞-connected (∞,1)-topos. There a converse to this conjecture is stated:

Propositon

Let CC be a site coming from a coverage such that constant (∞,1)-presheaves satisfy descent over objects of CC with respect to the generating covering families. Then the (∞,1)-category of (∞,1)-sheaves H=Sh (,1)(C)\mathbf{H} = Sh_{(\infty,1)}(C) is a locally ∞-connected (∞,1)-topos.

Revised on January 15, 2011 14:54:20 by Urs Schreiber (89.204.153.96)