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chain homotopy equivalence (Definition)

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.



"chain homotopy equivalence" is owned by CWoo.
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See Also: homotopy equivalence

Also defines:  chain homotopic equivalent
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Cross-references: equivalence relation, chain homotopy, chain map, morphism, chain complexes, abelian category, objects
There are 3 references to this entry.

This is version 3 of chain homotopy equivalence, born on 2004-11-24, modified 2004-11-24.
Object id is 6525, canonical name is ChainHomotopyEquivalence.
Accessed 2690 times total.

Classification:
AMS MSC: 18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes)
Pending Errata and Addenda
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