Topological half exact functor

In mathematics, a topological half-exact functor F is a functor from a fixed topological category (for example CW complexes or pointed spaces) to an abelian category (most frequently in applications, category of abelian groups or category of modules over a fixed ring) that has a following property: for each sequence of the form:

where C(f) denotes a mapping cone, a sequence:

is exact.

Examples include functors of homology.