Semiunital Semimonoidal Categories (applications to Semirings and Semicorings)

The category ASA of bisemimodules over a semialgebra A, with the so called Takahashi’s tensor-like product − A −, is semimonoidal but not monoidal. Although not a unit in ASA, the base semialgebra A has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we investigate semiunital semimonoidal categories (V, •, I) as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call J-monads (J-comonads) with respect to the endo-functor J := I•− ' −•I : V −→ V. This motivated also introducing a more generalized notion of monads (comonads) in arbitrary categories with respect to arbitrary endo-functors. Applications to the semiunital semimonoidal variety (ASA, A, A) provide us with examples of semiunital A-semirings (semicounital A-semicorings) and semiunitary semimodules (semicounitary semicomodules) which extend the classical notions of unital rings (counital corings) and unitary modules (counitary comodules).