# nLab contractible space

### Context

#### Topology

topology

algebraic topology

## Examples

#### Higher category theory

higher category theory

# Contents

## Definition

### General

An ∞-groupoid or a topological space or another realization of the concept (∞,0)-category is contractible if it is weakly equivalent to the point.

$\left(C\phantom{\rule{thickmathspace}{0ex}}\text{is contractible}\right)⇔\left(C\stackrel{\simeq }{\to }*\right)\phantom{\rule{thinmathspace}{0ex}}.$(C \;\text{is contractible}) \Leftrightarrow (C \stackrel{\simeq}{\to} *) \,.

Sometimes one allows also the empty object $\varnothing$ 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.

Notice that since the Whitehead theorem applies in ∞Grpd, being weakly equivalent to the point is the same as there being a contraction.

### For topological spaces

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)-truncated(-1)-groupoid/truth valueh-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoidh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-$\infty$-groupoid

Revised on October 23, 2012 21:17:59 by Urs Schreiber (131.174.191.164)