Paths and cylinders
Given an abelian category , write for its category of chain complexes.
for the category whose objects are chain complexes, and whose morphisms between objects are chain homotopy-equivalence classes
Analogous to the variants discussed at category of chain complexes one considers also the full subcategories
on the chain complexes which are bounded above or bounded below or bounded, respectively.
By construction there is a canonical functor
which is the identity? on objects and the quotient projection on hom-sets.
Additive category structure
In fact using the abelian group-structure on chain maps, we can equivalently reformulate the quotient by chain homotopy as follows:
For , there is an natural isomorphism of abelian groups
of the hom-objects in with the quotient group of all chain maps by those whose are null homotopic.
Triangulated category structure
A distinguished triangle in is a sequence of morphisms of the form
where denotes the suspension of a chain complex, which is isomorphic to the image under the projection fuctor def. 3 of a mapping cone sequence in
of a chain map in .
Relation to the derived category
See at derived category in the section derived category - Properties.
Chain homotopies that ought to exist but do not
We discuss some basic examples of chain maps that ought to be identified in homotopy theory, but which are not yet identified in , but only in the derived category .
In for Ab consider the chain map
The codomain of this map is an exact sequence, hence is quasi-isomorphic to the 0-chain complex. Thereofore in homotopy theory it should behave entirely as the 0-complex itself. In particular, every chain map to it should be chain homotopic to the zero morphism (have a null homotopy).
But the above chain map is chain homotopic precisely only to itself. This is because the degree-0 component of any chain homotopy out of this has to be a homomorphism of abelian groups , and this must be the 0-morphism, because is a free group, but is not.
This points to the problem: the components of the domain chain complex are not free enough to admit sufficiently many maps out of it.
Consider therefore a free resolution of the above domain complex by the quasi-isomorphism
where now the domain complex consists entirely of free groups. The composite of this with the original chain map is now
This is the corresponding resolution of the original chain map. And this indeed has a null homotopy:
Indeed, for this to happen it is sufficient that the resolution is by a degreewise projective complex. This is the statement of this lemma at projective resolution.
Lecture notes include
section 3.1 and 7.1 of