Natural numbers objects and free algebras

Natural numbers objects and free algebras

[Charles Wells]

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

Re: Natural numbers objects and free algebras

[S Vickers]

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.