nLab chain map

Idea

A chain map is a homomorphism of chain complexes. Chain complexes with chain maps between them form the category of chain complexes.

Definition

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).

Definition

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

$$\array{ V_{n+1} &\stackrel{d^V_n}{\to}& V_n \\ \downarrow^{\mathrlap{f_{n+1}}} && \downarrow^{\mathrlap{f_{n}}} \\ W_{n+1} &\stackrel{d^W_n}{\to} & W_n }$$

commute, hence such that all the equations

$$f_n \circ d^V_n = d^W_{n} \circ f_{n+1}$$

hold.

Remark

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.

Properties

On homology

Proposition

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

$$B_n(f) : B_n(C_\bullet) \to B_n(D_\bullet)$$

and

$$Z_n(f) : Z_n(C_\bullet) \to Z_n(D_\bullet) \,.$$

In particular it also respects chain homology

$$H_n(f) : H_n(C_\bullet) \to H_n(D_\bullet) \,.$$

Corollary

Conversely this means that taking chain homology is a functor

$$H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}$$

from the category of chain complexes in $\mathcal{A}$ to $\mathcal{A}$ itself.

In fact this is a universal delta-functor.

Related concepts

• category of chain complexes

  • chain complex

  • chain map, quasi-isomorphism

  • chain homotopy, null homotopy

References

A basic discussion is for instance in section 1.1 of

• Charles Weibel, An Introduction to Homological Algebra.

A more comprehensive discussion is in section 11 of

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves,