 Topological half-exact functor

For a contravariant pointed functor on based CW homotopy types, half-exactness on a cofibration means that $F(X/A)\to F(X)\to F(A)$ is exact at $F(X)$: 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.

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.