Semiunital Semimonoidal Categories (Applications to Semirings and Semicorings)

TL;DRAbstract

The category A S A 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 A S A , 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 ( A S A , 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

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