Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories


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



Paths and cylinders

Homotopy groups

Basic facts




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 understod as the abelianization of weak homotopy equivalences (see the Hurewicz theorem for more on this).


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


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.


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


Relation to chain homology type


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.


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}.


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

Relation to mapping cones and homotopy (co)fibers


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.


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.


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.


A basic introduction is around definition 1.1.2 in

A more systematic discussion is in section 12 of

Revised on January 17, 2015 14:55:56 by Adeel Khan (