categories: Characterizing FinSet up to equivalence

[baez@math.ucr.edu]

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