Homology of systemic modules

An informal overview of triples and systems

“…In [50] we also bring in tensor products and Hom functors. In [49] we develop the homological theory, relying on work done already by Grandis [27] under the name of N -category and homological category (without the negation map), and there is a parallel approach of Connes and Consani in [13].…”

“…In [50, §9], "Hom" is studied together with its dual, and again one only gets all the desired categorical properties (such as the adjoint isomorphism [50, Lemma 9.8]) when considering strict morphisms. In this way, the categories comprised of strict morphisms should be amenable to a categorical view, to be carried out in [49] for homology, again at times with a Hopfian flavor.…”

“…Building on projective modules, homology is work in progress [49], as is parallel work in geometry. One obtains a homology theory in the context of homological categories [27] and derived functors, in connection to the recent work of Connes and Consani [13].…”

“…It was motivated by an attempt to understand whether or not it is coincidental that basic algebraic theorems are mirrored in supertropical algebra, and was spurred by the realization that some of the same results are obtained in parallel research on hyperfields and fuzzy rings. Our objective is to hone in on the precise axioms that include these various examples, formulate the axiomatic structure, describe its uses, and review five papers [64,4,50,48,49] in which the theory is developed. The bulk of this survey concerns [64], in which the axiomatic framework is laid out, since the other papers build on it.…”

Morita theory of systems

“…We review some more definitions from [20,8,9,10] for the reader's convenience. A T -module is a T -module † with a distinguished element 0 ∈ A satisfying a0 A = 0 A for all a ∈ T .…”