Homotopy Category of Chain Complexes

The homotopy category of chain complexes K(A) is then defined as follows: its objects are the same as the objects of Kom(A), namely chain complexes. Its morphisms are "maps of complexes modulo homotopy": that is, we define an equivalence relation

if f is homotopic to g

and define

to be the quotient by this relation. It is clearer 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 (An=0 for n<<0), bounded-above (An=0 for n>>0), or bounded (An=0 for |n|>>0) complexes instead of unbounded ones, one speaks of the bounded-below homotopy category etc. They are denoted by K+(A), K-(A) and Kb(A), respectively.

A morphism which is an isomorphism in K(A) 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 (in the above sense) maps of singular chains.