nLab
locally contractible space

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)