As Fred says, the semimodules over a given semiring are determined by operations and equations, and so are complete and cocomplete. In terms of exactness properties they are also (i) locally finitely presentable (so that finite limits commute with filtered colimits, and (ii) Barr-exact (so that there is an equivalence between quotients and congruences) If we restrict to the cancellative case, we still have completeness and cocompleteness and (i), but (ii) fails. Steve Lack.
-----Original Message----- From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of Fred E.J. Linton Sent: Monday, March 17, 2008 4:43 AM To: categories@mta.ca Subject: categories: Re: 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