At 11:26 29/11/01 +0100, you wrote:
Hello,
the simplest characterization of FinSet known to me is as the free category with initial object and binary coproducts on one object. In the usual world existence of these types of coproducts is equivalent to finite coproducts, but restriction to nullary and binary ones avoids the need for an a priori notion of finiteness. Somehow this reminds me of Kuratowski's definition of finiteness.
Greetings Reinhard
No, I don't think you get the Kuratowski finite sets that way. A set X is Kuratowski finite iff it is in the subsemilattice (under nullary and binary union) of PX generated by the singletons. The definition proposed looks as though it characterizes finite ordinals, which are Kuratowski finite with a decidable total ordering. The two are different. "Kuratowski finite" includes sets where you can give a finite enumeration (indexed by a finite ordinal) of the elements but can't guarantee to eliminate duplicates from the enumeration. The category of Kuratowski finite sets is equivalent to the ind completion of the category of finite ordinals with surjections as the morphisms. Steve Vickers.