RE: Finiteness in Toposes

[categories <cat-dist@mta.ca>]

Date: Wed, 22 Jan 1997 18:18:53 +0000
From: Steve Vickers <sjv@doc.ic.ac.uk>

>Date: Fri, 17 Jan 1997 12:50:13 -0500 (EST)
>From: F William Lawvere <wlawvere@ACSU.Buffalo.EDU>
>
>Re:  Finiteness in Toposes                              Jan 17 1997
>
>This concerns the possibility , mentioned in my previous message, of two
>internal toposes of finite objects.
>
>        The conjecture that there are two natural internal categories of
>finite objects is partly supported by the fact that there are two natural
>natural-numbers objects, the usual one N  that parameterizes compositional
>iteration and another semicontinuous one L with the following features:
>
>...
>
>        This object L has been discussed for 25 years, but I dont know if
>anyone published the working-out of its properties and role.

Am I right in thinking this to be Idl N, the ideal completion of the
natural numbers (with their usual order)?

I conjecture that this is a suitable value domain for the ranks of matrices
over localic fields such as the reals: rank^-1{n} is not open, but
rank^-1{n, n+1, n+2, ...} is. Then rank A is the set of natural numbers n
such that we can find enough apartnesses to prove linear independence of n
rows of A, and this is an ideal of N - the definition also smoothly
incorporates infinite matrices.

(Perhaps this is just one of the things that have have been discussed for
25 years and I'm reiventing it.)

Anyway, I have investigated Idl N as a fixpoint object (in the sense of
Crole and Pitts) in the category of Grothendieck toposes (modulo
2-categorical niceties that I didn't investigate too closely) in a paper
"Topical Categories of Domains".

Steve Vickers.