Re : Free categories Toposes with free categories are precisely toposes with NNO: in particular TFAE: 1) NNO 2) the forgetful functor from the category of internal monoids is tripleable 3) the forgetful functor from the category of internal categories to the category of internal graphs is triplable The equivalence of 1) and 2) is Theorem 6.41 in Johnstone's Topos Theory. That 3) implies 2) is immediate. That 1) implies 3) follows from the fact that Finite Limit Theories (aka Essentially Algebraic Theories) have tripleable categories of models over any topos with NNO (I don't recall a reference on that last result, but I suspect it must be in Barr and Wells Toposes, Triples and Theories). Re: e-mail address As you can see from the header, my e-mail address is now dyetter@math.ksu.edu Best Thoughts, David Yetter +++++++++++++++++++++++++++++++++++++++