nLab
n-connected space

Idea

An $n$-connected space is a generalisation of the pattern:

• space,
• inhabited space,
• path-connected space,
• simply connected space,
• etc.

Definition

A topological space is $n$-connected or $n$-simply connected if its homotopy groups are trivial up to degree $n$.

Special cases

• Any space is $(-2)$-simply connected.

• A space is $(-1)$-simply connected precisely if it has an element; that is if it is inhabited.

• A space is $0$-simply connected precisely if it is path-connected.

• A space is $1$-simply connected precisely if it is simply connected.

• A space is $\mathrm{\infty }$-simply connected precisely if it is weakly contractible.

Terminology

The traditional terminology is ‘$n$-connected’, but this violates the rule that ‘$1$-foo’ should mean the same as ‘foo’. This can be fixed by saying ‘$n$-simply connected’ instead, which also has the advantage of stressing that we are extending the change from connected to simply connected spaces.

Properties

An $n$-connected topological space is precisely an n-connected object in the (∞,1)-topos ∞Gpd, presented by the model category Top of topological spaces.