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 spaces, of the form:
where "C(f)" denotes a mapping cone, the sequence:
is exact. If "F" is a contravariant functor, it is half-exact if for each sequence of spaces as above,
the sequence "F(C(f))" → "F(Y)" → "F(X)" is exact.
Homology is an example of a half-exact functor, and
cohomology (and generalized cohomology theories) are examples of contravariant half-exact functors.
If "B" is any fibrant topological space, the (representable) functor "F(X)=[X,B]" is half-exact.