# nLab contractible space

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## 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.

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+2$$n$-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-$n$-groupoid
h-level $\infty$untruncatedhomotopy 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.