Re: The Category of Semimodules over Semirings

[Jawad Abuhlail <abuhlail@kfupm.edu.sa>]

Dear Prof. Linton,
Many thanks for your comments about the existence of coequalizers in
categories of semimodules.

What I mentioned (that the category of left semimodules over an arbitrary
semiring has in general no coequalizers) was due to CONFUSION caused by the
way some results in "M. Takahashi, Completeness and $C$-co completeness of
the category of semimodules. Math. Sem. Notes Kobe Univ. 10 (1982), no. 2,
551--562." are stated.

In that paper, Takahashi proved that the category of left semimodules over
an arbitrary semiring has $c$-coequalizers and is $c$-cocomplete (where $c :
R-smod ---> C-R-smod$ denote the functor that assigns to each semimodule its
associated cancellative semimodule).

Indeed his proof does not exclude that this category has coequalizers as I
(apparently incorrectly) stated. For your convenience, I summarize what
Takahashi did in the above mentioned paper:

Denote the category of left semimodules over a semirings $R$ by $R-smod$ and
its full subcategory of cancellative semimodules by $C-R-smod$. Then
$R-smod$ has products and equalizers, whence complete. Let $c : R-smod --->
C-R-smod$ denote the functor that assigns to each semimodule its associated
cancellative semimodule. This functor is left adjoint to the embedding
functor $U : C-R-smod ---> R-smod$. Then $R-smod$ has coproducts and
$c$-coequalizers, whence $c$-cocomplete.

The confusion is caused by his statement that "$c$-cocompleteness is a
relaxation of cocompleteness" and the last Corollary in the paper, in which
he deduced that the full subcategory $C-R-smod$ of CANCELLATIVE semimodules
has coequalizers and is cocomplete!!

Anyway, I am so grateful for your comments and would appreciate as well any
comment about exactness of colimits in $C-R-smod$ and $C-R-smod$ (in the
category of modules over rings, colimits are exact!!) will be highly
appreciated.

Wassalam,
Jawad


-----Original Message-----
From: Fred E.J. Linton [mailto:fejlinton@usa.net]
Sent: Sunday, March 16, 2008 8:43 PM
To: categories@mta.ca
Cc: Jawad Abuhlail
Subject: Re: categories: The Category of Semimodules over Semirings

On Sun, 16 Mar 2008 09:26:35 AM EDT Jawad Abuhlail <abuhlail@kfupm.edu.sa>
wrote, in part, on the Subject: The Category of Semimodules over Semirings,

> ...  The category of semimodules had products, equalizers and products
> (however not necessarily coequalizers).

I must be missing something here. Don't the (say, left-) semimodules (over a
given semiring) constitute an equationally definable class of algebras? That
is, aren't they determined entirely by operations and equations?

If they DO, that is, if they ARE, then the category of them all (together
with their homomorphisms) must, like all such "varietal categories," have
all (small) limits and colimits, and, in particular, all coequalizers.

Alas, I have little else to offer. Cheers, and Happy St. Paddy's Day,

-- Fred