Editing Homology (mathematics) (section)

== Homology theories ==
To associate a ''homology theory'' to other types of mathematical objects, one first gives a prescription for associating chain complexes to that object, and then takes the homology of such a chain complex. For the homology theory to be valid, all such chain complexes associated to the same mathematical object must have the same homology. The resulting homology theory is often named according to the type of chain complex prescribed. For example, [[singular homology]], [[Morse homology]], [[Khovanov homology]], and [[Hochschild homology]] are respectively obtained from singular chain complexes, Morse complexes, Khovanov complexes, and Hochschild complexes. In other cases, such as for [[Group cohomology|group homology]], there are multiple common methods to compute the same homology groups.

In the language of [[category theory]], a homology theory is a type of [[functor]] from the [[Category (mathematics)|category]] of the mathematical object being studied to the category of abelian groups and group homomorphisms, or more generally to the category corresponding to the associated chain complexes. One can also formulate homology theories as [[Derived functor|derived functors]] on appropriate [[abelian category|abelian categories]], measuring the failure of an appropriate functor to be [[exact functor|exact]]. One can describe this latter construction explicitly in terms of [[Resolution (algebra)|resolutions]], or more abstractly from the perspective of [[Derived category|derived categories]] or [[Model category|model categories]].

Regardless of how they are formulated, homology theories help provide information about the structure of the mathematical objects to which they are associated, and can sometimes help distinguish different objects.