contractible space



Higher category theory

higher category theory

Basic concepts

Basic theorems





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 is 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

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

Revised on January 12, 2017 23:02:47 by Toby Bartels (