# nLab contractible space

### Context

#### Topology

topology

algebraic topology

## Examples

#### 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 is 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.

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 value(0,1)-sheafmere 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)-sheafh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheafh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoid(n+1,1)-sheafh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-$\infty$-groupoid

Revised on May 3, 2016 06:26:08 by Urs Schreiber (131.220.184.222)