Vaughan Pratt writes:
For ignorance of the correct name I'll call an object b "strongly indecomposable" when Hom(b,-) preserves binary sums.
I've heard this called "connected", which seems very nice, since that's what it amounts to in Top.
"Successor object" seems like a reasonable name for an object of the form b+1 (1 the final object). Write FinC for the full subcategory of C whose objects have finitely many elements (morphisms from 1).
Claim. Let C be a category with finite sums and final object 1. If 1 is a strongly indecomposable generator and every object is either initial or a successor, then FinC is equivalent to FinSet.
Since the concept of "finite set" is sitting right in the definition of FinC, we have to know all about finite sets to use this characterization of FinSet... but I wouldn't be surprised if that annoying circularity is inevitable. I wonder if anyone knows a reference to this characterization, which is simpler and perhaps more blatantly circular: Claim: FinSet is the free category with finite sums on one object. This is supposed to be a short way of saying that if C is a category with finite sums containing an object x, there is a finite-sum-preserving functor F: FinSet -> C, unique up to natural isomorphism, such that F(1) = x. It's a categorification of the fact that the natural numbers are the free commutative monoid on one generators. I think it's even true. I think this is also true: Claim: FinSet is the free biCartesian category on nothing. This is supposed to be a short way of saying that if C is a category with finite sums and finite products, the latter distributing over the former, then there is is a finite-sum-and- product-preserving functor F: FinSet -> C, unique up to natural isomorphism. It's a categorification of the fact that the natural numbers are the free commutative rig on no generators. Personally I find this sort of characterization a bit more illuminating than Vaughan's. Throughout math, as soon as you define some nice sort of gadget, you instantly focus on the free gadgets of this sort - which are probably the ones you knew about before you even made the definition! It's a circular business, but that's life. Best, John Baez