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.