Noncommutative Geometry through Monoidal Categories I

After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. In all these considerations we lay stress on the role of the monoidal structure, and the difference between this approach and the approach using (in general non-monoidal) abelian categories as models for categories of quasicoherent sheaves on noncommutative schemes.