This is a term which we invented for referring the general notion(s) in the book of Baues. I think he uses the term homology and cohomology of T-complexes. Special cases: Homology of groups, homology in a variety of groups, Hochschild homology of an algebra, homology of Lie algebras, homology of a topological space, Bredon homology of a G-space, homology theory for diagrams of spaces, controlled spaces, compactifications, and many other things.
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CT (Category theory)
Baues
nLab page on Baues cohomology