Toward weakly enriched categories: co-Segal categories

We introduce a new type of weakly enriched categories over a given symmetric monoidal model category MM; we call them co-Segal categories. Their definition derives from the philosophy of classical (enriched) Segal categories. The purpose of this paper is to expose the theory and give the first results on the homotopy theory of these structures. One of the motivations of developing such theory, was to have an alternative definition of higher linear categories following Segal-like methods.