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

An object $X$ of a cohesive $\infty$-topos may be contractible (or not) in different ways (modes):

• The object $X$ itself may be contractible in that it is (-2)-truncated. This is an extremely strong notion: It says that $X$ is the terminal object: the point, $X \simeq \ast$.

• The underlying $\infty$-groupoid $\flat X$ may be contractible, $\flat X \simeq \ast$.

  This condition is slightly weaker than full contractibility. For instance an infinitesimally thickened point $\mathbb{D}$, regarded as a 0-truncated haloed smooth $\infty$-groupoid, is generally not the actual point, but has $\flat \mathbb{D} \simeq \ast$.

• Its shape $\esh X \in Grpd_\infty$ may be contractible, $\esh X \simeq \ast$.

  This is the geometric (or cohesive) notion of contractibility.

  For instance a Cartesian space (vector space) $\mathbb{R}^n$, regarded as a 0-truncated smooth $\infty$-groupoid has contractible shape, $\esh \mathbb{R}^n \simeq \ast$.

  More generally, a topological space (or smooth manifold) $X$, has contractible shape when regarded as a 0-truncated Euclidean-topological $\infty$-groupoid (smooth $\infty$-groupoid) precisely if it is a weakly contractible topological space in the traditional sense (above).

• Its codiscrete aspect $\sharp X$ may be contractible, $\sharp X \simeq \ast$.

  But this is equivalent to $\flat X$ being contractible (since $\flat \sharp \simeq \flat$ and $\sharp \circ \flat \simeq \sharp$, and $\flat \ast \simeq \ast \simeq \sharp \ast$).

Examples

• An inhabited convex (or even just star-convex) subset of a topological vector space over $\mathbb{R}$ or $\mathbb{C}$ is contractible.

• The infinite-dimensional unit sphere in a separable infinite-dimensional Hilbert space is contractible, unlike the case of finite-dimensional spheres.

• By Kuiper's theorem the unitary group U(ℋ) of such a Hilbert space is contractible, where the topology can be either of the two main topologies (norm topology, or strong operator topology).

• The total space of any universal principal bundle is contractible.

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

References

See also:

• Ponaki Das, Sainkupar Marwein Mawiong: On Weakly Contractible Non-Contractible Finite Topological Spaces of Ten Points [arXiv:2605.06155]