categories: distributive(``bicartesian'' categories)

[Steve Lack <stevel@maths.usyd.edu.au>]

Toby Bartels writes:

 > John Baez wrote in part:
 > 
 > >Claim: FinSet is the free biCartesian category on nothing.
 > 
 > What is the justification for including in the term "biCartesian"
 > that the products distribute over the coproducts?
 > If you add that the Cartesian product is closed
 > (which it is in FinSet), *then* you get this, of course.
 > So FinSet is either the free biCartesian category
 > where products distribute over coproducts on nothing,
 > or else the free Cartesian closed coCartesian category on nothing.
 > It would be nice to have a single term like "biCartesian"
 > for either of these concepts, but I don't see the justification,
 > especially since the concept isn't very symmetric.
 > 

These categories are often called distributive. For an introduction
to them, and their relationship with extensive categories, see the
paper:

     Aurelio Carboni, Stephen Lack, and R.F.C. Walters, Introduction 
     to extensive and distributive categories, J. Pure Appl. Alg. 84(1993), 
     145-158.

Steve Lack.