Homotopy category of chain complexes


In homological algebra in mathematics, the homotopy category K of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom of A and the derived category D of A when A is abelian; unlike the former it is a triangulated category, and unlike the latter its formation does not require that A is abelian. Philosophically, while D makes isomorphisms of any maps of complexes that are quasi-isomorphisms in Kom, K does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, K is more understandable than D.

Definitions

Let A be an additive category. The homotopy category K is based on the following definition: if we have complexes A, B and maps f, g from A to B, a chain homotopy from f to g is a collection of maps such that
This can be depicted as:
We also say that f and g are chain homotopic, or that is null-homotopic or homotopic to 0. It is clear from the definition that the maps of complexes which are null-homotopic form a group under addition.
The homotopy category of chain complexes K is then defined as follows: its objects are the same as the objects of Kom, namely chain complexes. Its morphisms are "maps of complexes modulo homotopy": that is, we define an equivalence relation
and define
to be the quotient by this relation. It is clear that this results in an additive category if one notes that this is the same as taking the quotient by the subgroup of null-homotopic maps.
The following variants of the definition are also widely used: if one takes only bounded-below, bounded-above, or bounded complexes instead of unbounded ones, one speaks of the bounded-below homotopy category etc. They are denoted by K+, K− and Kb, respectively.
A morphism which is an isomorphism in K is called a homotopy equivalence. In detail, this means there is another map , such that the two compositions are homotopic to the identities: and
The name "homotopy" comes from the fact that homotopic maps of topological spaces induce homotopic maps of singular chains.

Remarks

Two chain homotopic maps f and g induce the same maps on homology because sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi-isomorphism. This shows that there is a canonical functor to the derived category.

The triangulated structure

The shift A of a complex A is the following complex
where the differential is.
For the cone of a morphism f we take the mapping cone. There are natural maps
This diagram is called a triangle. The homotopy category K is a triangulated category, if one defines distinguished triangles to be isomorphic to the triangles above, for arbitrary A, B and f. The same is true for the bounded variants K+, K− and Kb. Although triangles make sense in Kom as well, that category is not triangulated with respect to these distinguished triangles; for example,
is not distinguished since the cone of the identity map is not isomorphic to the complex 0. Furthermore, the rotation of a distinguished triangle is obviously not distinguished in Kom, but is distinguished in K. See the references for details.

Generalization

More generally, the homotopy category Ho of a differential graded category C is defined to have the same objects as C, but morphisms are defined by
.. If C has cones and shifts in a suitable sense, then Ho is a triangulated category, too.