nLab chain map

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

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. 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

Last revised on October 2, 2019 at 09:24:38. See the history of this page for a list of all contributions to it.