Peter, These are pretty wild questions!!! :-) I am presuming you know that FSet^Dop is not the finite cocompletion as in general D does not embedd in this!! If D is a finite category you are home and dry of course. But this is just the same thing as assuming that the universe is finite sets. The free completion with respect to coproducts is the "Fam" construction. I took a look at this in my "Introduction to distributive categories" MSCS 1991 (see towards the end): Steve Schanuel has recently been revisiting this (at the Union meeting). -robin On 24 Nov, Peter Selinger wrote:
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