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# chain homotopy equivalence

Let $C$ and $D$ be two objects from the abelian category of chain complexes. A morphism (or chain map) $f\colon C\to D$ is said to be a *chain homotopy equivalence* if there is a morphism $g\colon D\to C$ such that

1. there is a chain homotopy between $fg$ and $1\colon D\to D$; and

2. there is a chain homotopy between $gf$ and $1\colon C\to C$.

If a chain homotopy equivalence from a chain complex $C$ to $D$ exists, then $C$ is said to be *chain homotopy equivalent* to $D$. Chain homotopy equivalence is an equivalence relation among chain complexes.

Defines:

chain homotopic equivalent

Related:

HomotopyEquivalence

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

18G35*no label found*

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