nLab contractible space

Definition

For topological spaces

A topological space $X$ is contractible if the canonical map $X \to \ast$ is a homotopy equivalence. It is weakly contractible if this map is a weak homotopy equivalence, hence if all homotopy groups of $X$ are trivial.

Where the Whitehead theorem does not apply, we may find examples of weakly contractible but not contractible spaces, such as the double comb space in Top.

For $\infty$-groupoids

Since the Whitehead theorem applies in ∞Grpd (and generally in any hypercomplete (∞,1)-topos), being weakly equivalent to the point is the same as there being a contraction. So an ∞-groupoid is weakly contractible if and only if it is contractible.

$$(C \;\text{is weakly contractible}) \Leftrightarrow (C \stackrel{\simeq}{\to} *) \,.$$

In this context one tends to drop the “weakly” qualifier.

Sometimes one allows also the empty object $\emptyset$ to be contractible. To distinguish this, we say

• an $\infty$-groupoid is (-1)-truncated (is a (-1)-groupoid) if it is either empty or equivalent to the point;

• an $\infty$-groupoid is (-2)-truncated (is a (-2)-groupoid) if it is equivalent to the point.

For cohesive $\infty$-groupoids

Cohesive $\infty$-groupoids could be contractible in two different ways: topologically contractible in the first sense, or homotopically contractible in the second sense. A cohesive $\infty$-groupoid $S$ is homotopically contractible if its underlying $\infty$-groupoid $\Gamma(S)$ is contractible. A cohesive $\infty$-groupoid is topologically contractible if its fundamental infinity-groupoid $\Pi(S)$ is contractible. These two notions of contractibility are not equivalent to each other: in Euclidean-topological infinity-groupoids the unit interval is topologically contractible, but homotopically the unit interval is only 0-truncated.

Related concepts

• locally contractible space

• contractible chain complex

• contractible type

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/​unit type/​contractible type
h-level 1	(-1)-truncated	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/​set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/​groupoid	(2,1)-sheaf/​stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-$\infty$-groupoid