nLab quasi-isomorphism

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

Definition

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

$$H_n(f) \colon H_n(C) \to H_n(D)$$

is an isomorphism.

Remark

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

Properties

Relation to chain homology type

Proposition

The relation “There exists a quasi-isomorphism from $C_\bullet$ to $D_\bullet$.” 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

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

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 $\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}$.

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

Proof

This follows for instance from the homology long exact sequence

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

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

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

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

Related concepts

• weak homotopy equivalence

• injective resolution, projective resolution

• category of chain complexes

  • chain complex

  • chain map

  • chain homotopy

References

A basic introduction is around definition 1.1.2 in

• Charles Weibel, An Introduction to Homological Algebra

A more systematic discussion is in section 12 of

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves