Homology with models and Tor (Q1360362)

The notion of projective functor from a category \(\mathcal C\) to \({\mathcal A}b\) (the category of abelian groups) relative to a model set is an extension of the notion of projective module. A good reference is: \textit{A. Dold}, \textit{S. MacLane} and \textit{U. Oberst} [``Projective classes and acyclic models'', in: Reports of the Midwest Category Seminar, Lect. Notes Math. 47, 78-91 (1967; Zbl 0165.33001)]. Taking projective resolutions in this context, one has an associated homology. Giving the term of \({\mathcal C}\)-module and the resolution of such, one has the framework of \textit{G. Hoff} [``On the cohomology of categories'', Rend. Mat., VI. Ser. 7, 169-192 (1974; Zbl 0361.18012)] and others. Studying change of model sets, the authors show how the model set can be replaced by a model set having only one element. Then, the homology modules can be interpreted in terms of torsion functors. Very nice examples are the cases of classical simplicial homology of topological spaces and of artinian modules over a commutative local ring.