nLab quasi-isomorphism

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

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

A quasi-isomorphism is a chain map that induces isomorphisms on all homology groups. These are the natural choice of weak equivalences between chain complexes in the context of (stable) homotopy theory.

The localization of a category of chain complexes at the quasi-isomorphisms is called the derived category of the underlying abelian category.

Under the relation between topological spaces and chain complexes established by forming singular simplicial complexes, quasi-isomorphism can be understood as the abelianization of weak homotopy equivalences (see the Hurewicz theorem for more on this).

Definition

Let 𝒜\mathcal{A} be an abelian category and write Ch (𝒜)Ch_\bullet(\mathcal{A}) for its category of chain complexes.

Definition

A chain map f :C D f_\bullet : C_\bullet \to D_\bullet in Ch (𝒜)Ch_\bullet(\mathcal{A}) is called a quasi-isomorphism if for each nn \in \mathbb{N} the induced morphisms on chain homology groups

H n(f):H n(C)H n(D) H_n(f) \colon H_n(C) \to H_n(D)

is an isomorphism.

Remark

Quasi-isomorphisms are also called, more descriptively, homology isomorphisms or H H_\bullet-isomorphisms. See at homology localization for more on this.

Properties

Relation to chain homology type

Proposition

The relation “There exists a quasi-isomorphism from C C_\bullet to D D_\bullet.” is a reflexive and transitive relation, but it is not a symmetric relation.

Proof

Reflexivity and transitivity are evident. An explicit counter-example showing the non-symmetry is the chain map

0 2 0 0 0 /2 0 \array{ \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\to& 0 &\to& \cdots \\ \cdots && \downarrow && \downarrow && \downarrow && \downarrow && \cdots \\ \cdots &\to& 0 &\to& 0 &\to& \mathbb{Z}/2\mathbb{Z} &\to& 0 &\to& \cdots }

from the chain complex concentrated on the morphism of multiplication by 2 on integers, to the chain complex concentrated on the cyclic group of order 2.

This clearly induces an isomorphism on all homology groups. But there is not even a non-zero chain map in the other direction, since there is no non-zero group homomorphism /2\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}.

Remark

This is as for weak homotopy equivalences, see the discussion at Relation to homotopy types there.

Relation to mapping cones and homotopy (co)fibers

Proposition

A chain map is a quasi-isomorphism precisely if its homotopy cofiber in the (∞,1)-category of chain complexes has trivial homology groups.

By basic properties discussed at truncated object in an (∞,1)-category.

Concretely this means in particular the following.

Proposition

A chain map f :C D f_\bullet : C_\bullet \to D_\bullet is a quasi-isomorphism precisely if its mapping cone cone(f) Ch (𝒜)cone(f)_\bullet \in Ch_\bullet(\mathcal{A}) has all trivial chain homology groups.

Proof

This follows for instance from the homology long exact sequence

H n+1(c)H n+1(D)H n+1(cone(f))H n(C)H n(D)H n(cone(f))H n1(C)H n1(D)H n1(cone(f)). \cdots \to H_{n+1}(c)\to H_{n+1}(D) \to H_{n+1}(cone(f)) \to H_n(C) \to H_n(D) \to H_n(cone(f)) \to H_{n-1}(C) \to H_{n-1}(D) \to H_{n-1}(cone(f)) \to \cdots \,.

If here by assumption H n(cone(f))=0H_n(cone(f)) = 0 for all nn, then this involves exact sequences of the form

0H n(C)H n(f)H n(D)0 0 \to H_n(C) \stackrel{H_n(f)}{\to} H_n(D) \to 0

for all nn. But this says that the kernel and cokernel of H n(f)H_n(f) are trivial for all nn, hence that H n(f)H_n(f) is an isomorphism for all nn, hence that f f_\bullet is a quasi-isomorphism.

In homotopy theory

Quasi-isomorphisms are the weak equivalences in the most common model category structures on the category of chain complexes. See at model structure on chain complexes and derived category.

References

A basic introduction is around definition 1.1.2 in

A more systematic discussion is in section 12 of

Last revised on February 3, 2019 at 20:46:33. See the history of this page for a list of all contributions to it.