(also nonabelian homological algebra)
A chain map is a homomorphism of chain complexes. Chain complexes with chain maps between them form the category of chain complexes.
Let $V_\bullet, W_\bullet \in Ch_\bullet(\mathcal{A})$ be two chain complexes in some ambient additive category $\mathcal{A}$ (often assumed to be an abelian category).
A chain map $f : V_\bullet \to W_\bullet$ is a collection of morphism $\{f_n : V_n \to W_n\}_{n \in \mathbb{Z}}$ in $\mathcal{A}$ such that all the diagrams
commute, hence such that all the equations
hold.
A chain map $f$ induces for each $n \in \mathbb{Z}$ a morphism $H_n(f)$ on homology groups, see prop. below. If these are all isomorphisms, then $f$ is called a quasi-isomorphism.
For $f : C_\bullet \to D_\bullet$ a chain map, it respects boundaries and cycles, so that for all $n \in \mathbb{Z}$ 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 $\mathcal{A}$ to $\mathcal{A}$ 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 05:24:38. See the history of this page for a list of all contributions to it.