nLab
contractible space

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higher category theory

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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.

(Cis contractible)(C*). (C \;\text{is contractible}) \Leftrightarrow (C \stackrel{\simeq}{\to} *) \,.

Sometimes one allows also the empty object \emptyset to be contractible. To distinguish this, we say

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 valuemere proposition, h-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+2n+2nn-truncatedhomotopy n-typen-groupoidh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-\infty-groupoid

Revised on July 1, 2013 05:59:09 by Toby Bartels (98.19.44.54)