nLab contractible space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Higher category theory

higher category theory

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Universal constructions

Extra properties and structure

1-categorical presentations



For topological spaces

A topological space XX is contractible if the canonical map X*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 XX 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.

(Cis weakly contractible)(C*). (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

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 SS is homotopically contractible if its underlying \infty-groupoid Γ(S)\Gamma(S) is contractible. A cohesive \infty-groupoid is topologically contractible if its fundamental infinity-groupoid Π(S)\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.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid

Last revised on June 18, 2022 at 15:51:49. See the history of this page for a list of all contributions to it.