Paths and cylinders
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 be an abelian category and write for its category of chain complexes.
A chain map in is called a quasi-isomorphism if for each the induced morphisms on chain homology groups
is an isomorphism.
Relation to chain homology type
Reflexivity and transitivity are evident. An explicit counter-example showing the non-symmetry is the chain map
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 .
Relation to mapping cones and homotopy (co)fibers
By basic properties discussed at truncated object in an (∞,1)-category.
Concretely this means in particular the following.
A chain map is a quasi-isomorphism precisely if its mapping cone has all trivial chain homology groups.
This follows for instance from the homology long exact sequence
If here by assumption for all , then this involves exact sequences of the form
for all . But this says that the kernel and cokernel of are trivial for all , hence that is an isomorphism for all , hence that 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