Homology (mathematics) - Homology Functors

Homology Functors

Chain complexes form a category: A morphism from the chain complex (d_n: A_n → A_n-1) to the chain complex (e_n: B_n → B_n-1) is a sequence of homomorphisms f_n: A_n → B_n such that for all n. The n-th homology H_n can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups (or modules).

If the chain complex depends on the object X in a covariant manner (meaning that any morphism X → Y induces a morphism from the chain complex of X to the chain complex of Y), then the H_n are covariant functors from the category that X belongs to into the category of abelian groups (or modules).

The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups (which are called cohomology groups in this context and denoted by Hn) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.