Horizontal categorification of spectral triples

We study duality for multimodules and we also describe further notions on multimodules: generalized morphisms and involutions; tensor products and contractions; inner products; first-order differential operators. We use the notion of noncommutative vector fields by A. Borowiec to define a left/right derivations on noncommutative algebras and we give in the involutive case a conjugation between right and left dualities of covariant and contravariant differential calculi. Finally we introduce definitions of spectral triples over a C*-category and provide some examples.