nLab
chain map

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

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 Ch (𝒜)V_\bullet, W_\bullet \in Ch_\bullet(\mathcal{A}) be two chain complexes in some ambient additive category 𝒜\mathcal{A} (often assumed to be an abelian category).

Definition

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

V n+1 d n V V n f n+1 f n W n+1 d n W W n \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 nd n V=d n Wf n+1 f_n \circ d^V_n = d^W_{n} \circ f_{n+1}

hold.

Remark

A chain map ff induces for each nn \in \mathbb{Z} a morphism H n(f)H_n(f) on homology groups, see prop. 1 below. If these are all isomorphisms, then ff is called a quasi-isomorphism.

Properties

On homology

Proposition

For f:C D f : C_\bullet \to D_\bullet a chain map, it respects boundaries and cycles, so that for all nn \in \mathbb{Z} it restricts to a morphism

B n(f):B n(C )B n(D ) B_n(f) : B_n(C_\bullet) \to B_n(D_\bullet)

and

Z n(f):Z n(C )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 )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():Ch (𝒜)𝒜 H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}

from the category of chain complexes in 𝒜\mathcal{A} to 𝒜\mathcal{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 June 20, 2014 03:26:18 by Jan Pulmann? (78.128.168.45)