Greetings. On Fri, 15 Feb 2008 02:30:45 PM EST, "Jamie Vicary" <jamie.vicary@imperial.ac.uk> wrote:

Given a commutative semiring R, construct the category that has nonnegative integers as objects, and matrices with entries in R as morphisms, such that the hom-set from N to M consists of matrices with M rows and N columns. This is then just the category of finitely-generated free R-modules. It admits a symmetric monoidal structure given by tensor product of modules, and has finite biproducts, with the induced CMon-enrichment agreeing with addition in R. Is there a nice way to characterise the semirings R for which this category has binary equalisers, and hence all finite limits?

Note that the category is equal to its opposite, identifying each

matrix

with its transpose, and so it will have finite colimits iff it has finite limits. ...

While I offer no answer, I will point out that coequalizers here (and, by the same token, equalizers) need not look like what you might expect. For example, when the commutative semiring R is the ordinary ring of integers, so that the category of R-matrices "is" the full category of f.-g. free abelian groups, the coequalizer of the pair of 1x1 matrices (2): 1 --> 1 and (0): 1 --> 1 exists and is !: 1-->0 , with the "expected" coequalizer value of Z/2Z being "unavailable here." So: beware! Cheers, -- Fred