#
Semiunital semimonoidal categories (Applications to semirings and
semicorings)

##
Jawad Abuhlail

The category $_{A}\mathbb{S}_{A}$ of bisemimodules over a semialgebra $A,$
with the so called *Takahashi's tensor-like 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* $(\mathcal{V}% ,\bullet ,\mathbf{I})$ 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).

Keywords:
Semimonoidal Categories, Semiunits, Monads, Comonads, Semirings,
Semimodules, Semicorings, Semicomodules

2010 MSC:
18C15, 18D10, 16W30

*Theory and Applications of Categories,*
Vol. 28, 2013,
No. 4, pp 123-149.

Published 2013-02-26.

http://www.tac.mta.ca/tac/volumes/28/4/28-04.dvi

http://www.tac.mta.ca/tac/volumes/28/4/28-04.ps

http://www.tac.mta.ca/tac/volumes/28/4/28-04.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/4/28-04.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/4/28-04.ps

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