(also nonabelian homological algebra)
A chain complex $V$ is called contractible if the unique terminal morphism $V \to 0$ to the zero complex is a homotopy equivalence, hence if the identity morphism on $V$ is null homotopic. One also says that $V$ is null-homotopic.
It is called weakly contractible if $V \to 0$ is a quasi-isomorphism.
Every mapping cone on an identity chain map is contractible.
In quantization of Yang-Mills theory gauge fixing is typically implementd in BV-BRST formalism after introducing a contractible chain complex of field bundles for fields called the Nakanishi-Lautrup field and the antighost field.
homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|
h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |
h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |
h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |
h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |
h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
Last revised on October 12, 2017 at 15:32:04. See the history of this page for a list of all contributions to it.