nLab contractible chain complex

Definition

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.

Examples

• 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.

Related concepts

• contractible space

• contractible type

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