Date: Wed, 14 Oct 1998 12:10:39 -0400 From: Charles Wells Subject: categories: Natural numbers objects and free algebras 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. Thanks, Charles Wells, Department of Mathematics, Case Western Reserve University, 10900 Euclid Ave., Cleveland, OH 44106-7058, USA. EMAIL: charles@freude.com. OFFICE PHONE: 216 368 2893. FAX: 216 368 5163. HOME PHONE: 440 774 1926. HOME PAGE: URL http://www.cwru.edu/artsci/math/wells/home.html Date: Thu, 15 Oct 1998 09:42:56 +0100 From: S Vickers Subject: Re: categories: Natural numbers objects and free algebras 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.