categories: Category Theory and Hereditarily-Finite Sets

[Galchin Vasili <vngalchin@yahoo.com>]

Hello Cat Theory Community,

     Hereditarily-finite sets are becoming increasingly more popular in
computer science research.

1) What kind of interesting categories exist where an 
     "object" is a hereditarily-finite set plus some 
     structure on the set and a "morphism" would be a
     structure-preserving function. (I can think
     of the obvious subcategory of SET and also the 
     category where an object is a hereditarily-finite
     set together with a "SET" endomorphism on that 
     set, but neither of these categories would have 
     interesting or useful properties, in my opinion)
     

2) What kind of papers can I read on this subject?

Regards,

Bill Halchin