Subject: RE: Finiteness in Toposes

From: categories <[email-masked]>
Date: Fri, 17 Jan 1997 16:04:23 -0400 (AST)

Date: Fri, 17 Jan 1997 12:50:13 -0500 (EST)
From: F William Lawvere <[email-masked]>

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:

0)  It is a rig, so receives a homomorphism from N  and its
elementary arithmetic starts out looking very similar.

1)  But unlike N it has a least-number-property in the sense that it is
inf-complete and better.

2)  It can be constructed internally using truth-valued sheaves on N.

3)  Hence it also contains a map from (big) omega, which permits (unlike
N) the use of the standard method in finite combinatorics where (for
example) a binary relation is considered as a matrix which is valued (not
only in a rig where 1+1 = 1, but instead) in a rig in which natural
numbers are distinct;  the resulting generalized characteristic functions
are added, multiplied, infed etc. according to the usual methods of
arithmetic and analysis and then translated back into the combinatorics of
the original finite structures.  Of course, in each case one hopes that
the answer to a combinatorial problem might turn out decidable, but that
shouldnt require us to stay in the bounds of two-valued subsets in the
course of a construction.

4)  This internally-defined order-complete rig in E has also an external
characterization if E is an S-based topos, namely it is the sheaf of germs
of S-geometrical morphisms from E to the topos often called  S-sets
-through-time (I dont think that depends on any presumption that the N in
S ,used to parameterize the transitions through time, coincides with its
completion in S).  In localic or open set terms, there is in S a (T sub
zero) space whose points are N, but whose open sets have the usual order
on N as their specialization order;  continuous functions from any space E
to this space are called semi-continuous and there is in E a sheaf of
them.

5)  The application to the variable linear algebra over algebraic or
complex-analytic spaces needs L too, because dimension of a vector space
is a semi-continuous function.  More precisely, if  A  is a good module in
a ringed topos E, R then for each X and E there should be a map X--> L
which is the fiber-wise dimension of X*A.  The basic case is perhaps that
where E,R is an algebraic affine scheme, and the conceptual problem is to
get at what sort of sets contained in A this dimension function is
counting (or bounding). One should not expect that equality of dimension
will imply isomorphism.

	This object L has been discussed for 25 years, but I dont know if
anyone published the working-out of its properties and role.

Bill

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From: categories <[email-masked]>
Date: Wed, 22 Jan 1997 14:41:21 -0400 (AST)

Date: Wed, 22 Jan 1997 18:18:53 +0000
From: Steve Vickers <[email-masked]>

>...

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.