(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
A chain map is a homomorphism of chain complexes. Chain complexes with chain maps between them form the category of chain complexes.
Let be two chain complexes in some ambient additive category (often assumed to be an abelian category).
A chain map is a collection of morphism in such that all the diagrams
commute, hence such that all the equations
hold.
A chain map induces for each a morphism on homology groups, see prop. below. If these are all isomorphisms, then is called a quasi-isomorphism.
For a chain map, it respects boundaries and cycles, so that for all it restricts to a morphism
and
In particular it also respects chain homology
Conversely this means that taking chain homology is a functor
from the category of chain complexes in to itself.
In fact this is a universal delta-functor.
chain map, quasi-isomorphism
A basic discussion is for instance in section 1.1 of
A more comprehensive discussion is in section 11 of
Last revised on October 2, 2019 at 09:24:38. See the history of this page for a list of all contributions to it.