nLab locally contractible space

Definition

Definition

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

Sometimes one requires just that the inclusions $U \to X$ 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 $U$ have (just) trivial homotopy groups, but this does not seem to crop up in practice.

Definition

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

Examples

• Any CW-complex is locally contractible (see there).

• Any paracompact manifold is locally contractible.

• Any contractible space is semi-locally contractible

• The cone on the Hawaiian earring space is contractible, hence semi-locally contractible, but is not locally contractible, as any neighbourhood of the ‘bad point’ is not simply connected.

Properties

Proposition

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

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

Other viewpoints

If one considers fundamental ∞-groupoids, the inclusion $U \to X$ being null-homotopic is equivalent to the induced (∞,1)-functor $\Pi(U) \to \Pi(X)$ 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 $X$ is semi-locally contractible then every locally constant $n$-stack on the site of open sets of $X$ is locally trivial.

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

Propositon

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