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 $V_{•},W_{•}\in {\mathrm{Ch}}_{•}(𝒜)$ be two chain complexes in some ambient additive category $𝒜$ (often assumed to be an abelian category).

Definition

A chain map $f:V_{•}\to W_{•}$ is a collection of morphism $\{f_n:V_n\to W_n{\}}_{n\in \mathbb{Z}}$ in $𝒜$ such that all the diagrams

$$\begin{array}{ccc}{V}_{n+1}& \stackrel{d_n^V}{\to }& V_n\\ {\downarrow }^{f_{n+1}}& & {\downarrow }^{f_n}\\ W_{n+1}& \stackrel{d_n^W}{\to }& W_n\end{array}$$\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

$$f_n\circ d_n^V=d_{n+1}^V\circ f_{n+1}$$f_n \circ d^V_n = d^V_{n+1} \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. 1 below. If these are all isomorphisms, then $f$ is called a quasi-isomorphism.

Properties

On homology

Proposition

For $f:C_{•}\to D_{•}$ a chain map, it respects boundaries and cycles, so that for all $n\in \mathbb{Z}$ it restricts to a morphism

$$B_n(f):B_n(C_{•})\to B_n(D_{•})$$B_n(f) : B_n(C_\bullet) \to B_n(D_\bullet)

and

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

In particular it also respects chain homology

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

Corollary

Conversely this means that taking chain homology is a functor

$$H_n(-):{\mathrm{Ch}}_{•}(𝒜)\to 𝒜$$H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}

from the category of chain complexes in $𝒜$ to $𝒜$ 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,