Dear category theorists, 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. Also, we assume that our semiring has a 0 and a 1, is distributive, and that x.0=0.x=0 for all x in R. Jamie. PS: There was a interesting discussion of this towards the end of this page: http://golem.ph.utexas.edu/category/2007/11/geometric_representation_theor_1...