At 12:10 14/10/98 -0400, Charles Wells wrote:

I am vaguely aware that there are theorems to the effect that in a topos (pretopos?) with nno you have free algebras in some sense. Where can I read about that? What I want to know is what role the nno plays.

Johnstone and Wraith "Algebraic Theories in a Topos". The nno provides internally an infinite object. Once it is there, the elementary topos structure can be used to construct other free algebras. However, the constructions use exponentiation and subobject classifier, and pretopos structure is not enough. Adam Eppendahl and I are working on a conjecture that the structure of Joyal's Arithmetic Universes is sufficient to construct general free algebras, that structure comprising (following ideas of Joyal and Wraith) pretopos structure + free categories over graphs + free category actions over graph actions. Steve Vickers.