baez@math.ucr.edu wrote:
Claim: FinSet is the free category with finite sums on one object.
I wonder what happens in the case of more than one generator. For instance, the free category with finite sums on two objects is FinSet x FinSet. In the case where the set of generators is discrete, it does not make a difference if one also adds coequalizers, e.g. FinSet is the free category with finite colimits on one object. What about the case where one has morphisms on the generators? From [Mac Lane], we know: If D is any diagram (small category), then its free co-completion is the Yoneda category Set^{D^op}. Is this still true when inserting the word "finite"? If D is any diagram, then its free completion under finite colimits is FinSet^{D^op}? And what happens if one drops the coequalizers? Does the free completion of a diagram D under coproducts have a Yoneda-like characterization? -- Peter