# nLab chain map

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## 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 ${V}_{•},{W}_{•}\in {\mathrm{Ch}}_{•}\left(𝒜\right)$ 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 $\left\{{f}_{n}:{V}_{n}\to {W}_{n}{\right\}}_{n\in ℤ}$ in $𝒜$ such that all the diagrams

$\begin{array}{ccc}{V}_{n+1}& \stackrel{{d}_{n}^{V}}{\to }& {V}_{n}\\ {↓}^{{f}_{n+1}}& & {↓}^{{f}_{n}}\\ {W}_{n+1}& \stackrel{{d}_{n}^{W}}{\to }& {W}_{n}\end{array}$\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 $n\in ℤ$ a morphism ${H}_{n}\left(f\right)$ 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 $n\in ℤ$ it restricts to a morphism

${B}_{n}\left(f\right):{B}_{n}\left({C}_{•}\right)\to {B}_{n}\left({D}_{•}\right)$B_n(f) : B_n(C_\bullet) \to B_n(D_\bullet)

and

${Z}_{n}\left(f\right):{Z}_{n}\left({C}_{•}\right)\to {Z}_{n}\left({D}_{•}\right)\phantom{\rule{thinmathspace}{0ex}}.$Z_n(f) : Z_n(C_\bullet) \to Z_n(D_\bullet) \,.

In particular it also respects chain homology

${H}_{n}\left(f\right):{H}_{n}\left({C}_{•}\right)\to {H}_{n}\left({D}_{•}\right)\phantom{\rule{thinmathspace}{0ex}}.$H_n(f) : H_n(C_\bullet) \to H_n(D_\bullet) \,.
###### Corollary

Conversely this means that taking chain homology is a functor

${H}_{n}\left(-\right):{\mathrm{Ch}}_{•}\left(𝒜\right)\to 𝒜$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.

## References

A basic discussion is for instance in section 1.1 of

A more comprehensive discussion is in section 11 of

Revised on September 2, 2012 20:28:57 by Urs Schreiber (89.204.139.178)