For a contravariant pointed functor on based CW homotopy types, half-exactness on a cofibration means that is exact at : the image is the inverse image of the distinguished point. This does not demand injectivity or surjectivity at the ends. Set-valued exactness and group-valued exactness must be distinguished.
A reduced contravariant homotopy functor on based CW complexes satisfying the wedge-to-product axiom and the Mayer-Vietoris weak-pullback gluing axiom is represented by for a CW complex . Half-exactness alone is not the full hypothesis. A reduced generalized cohomology theory supplies these axioms; suspension compatibility assembles its representing spaces into an Omega-spectrum.
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In the context of category theory and algebraic topology, a topological half-exact functor is a type of functor that reflects certain properties related to homotopy and convergence, particularly in the context of topological spaces, simplicial sets, or other similar structures. While the term "topological half-exact functor" is not widely standardized or commonly used in the literature, it's likely referring to concepts related to exactness in categorical contexts.