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