Homology (mathematics)

In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). See homology theory for more background.

Homology functors

Chain complexes form a category: A morphism from the chain complex (dn : An -> An-1) to the chain complex (en : Bn -> Bn-1) is a sequence of homomorphisms fn : An -> Bn such that fn-1 o dn = en-1 o fn for all n. The n-th homology Hn can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups (or modules).

Related Topics:
Category - Functor

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

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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.

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