Semiunital Semimonoidal Categories (Applications to Semirings and Semicorings)

TL;DR

This paper explores semiunital semimonoidal categories, introducing new notions of semimonoids, monads, and their applications to semirings and semicorings, extending classical algebraic concepts.

Contribution

It develops the theory of semiunital semimonoidal categories and introduces generalized monads and comonads, with applications to semirings and semicorings.

Findings

Examples of semiunital A-semirings and semicorings provided.

Extension of classical unital and counital notions to semiunital context.

Framework for studying semimonoids and related structures in semiunital categories.

Abstract

The category $_{A}\mathbb{S}_{A}$ of bisemimodules over a semialgebra $A,$ with the so called Takahashi's tensor product $-\boxtimes_{A}-,$ is semimonoidal but not monoidal. Although not a unit in $_{A}\mathbb{S}%_{A},$ 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 $(\mathcal{V}%, \bullet, I)$ as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call $\mathbb{J}$-monads ($\mathbb{J}$-% comonads) with respect to the endo-functor $\mathbb{J}:=\mathbf{I}\bullet -\simeq -\bullet \mathbf{I}:\mathcal{V}\longrightarrow \mathcal{V}.$ 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 $(_{A}\mathbb{S}%_{A},\boxtimes_{A},A)$ 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).