finiteness

finiteness

[categories]

Date: Sun, 26 Jan 1997 18:16:28 EST
From: carboni@vmimat.mat.unimi.it

Regarding my last message on finiteness, I should have said `functors
N---->Omega' instead of `ideals N--->Omega'. I repeat that I `think' that
this is what Bill wanted to say, but I am not sure that I am correct.

As for the reference of Bill's Perugia Notes, they are an internal publication
of Perugia University in 1972 of the lectures given by Bill Lawvere when he
was visiting that University. They should be available there (write to prof.
L. Stramaccia, Dipartimento di Matematica, Universita' di Perugia,
via Vanvitelli 1, 06123 Perugia, Italy, email: stra@gauss.dipmat.unipg.it).
Also, they were quite spread out, so that you should be able to find somebody
nearby you who has them. Other possibilities are asking the author himself and
eventually myself.

                        Aurelio Carboni.

Re: finiteness

[categories]

Date: Fri, 24 Jan 1997 10:54:59 +0000
From: Steve Vickers <sjv@doc.ic.ac.uk>

>From: carboni@vmimat.mat.unimi.it
>
>... L consists of those ideals T : N---->Omega such that for all n
>Tn = Inf{Tm | m > n}.

I understand this as saying that n is in the ideal iff every greater m is
in the ideal (but I think the inequality m > n has to be non-strict to make
sense of this). Hence it's really a filter of N.

If that's correct, then my suggestion was wrong. L would be not Idl N, but
Idl(N^op). That makes sense regarding dimensions, for if a real vector
space is finitely presented using an mxn matrix A (presenting R^n/Im A)
then its dimension is n-rank(A), so if rank(A) is in Idl(N), the dimension
should be in Idl(N^op).

(By the way, what's a full reference for the "Perugia Notes"?)

Steve.

finiteness

[categories]

Date: Thu, 23 Jan 1997 19:43:40 EST
From: carboni@vmimat.mat.unimi.it

Regarding the last message of Bill on finiteness and the answer of Vickers, I
would like to point out that the 25 years old reference of Bill should be the
one at the bottom of page 14 of lesson 3 of 1972 Perugia Notes. In other words,
the L he is suggesting should be the internalization of the following
description: L consists of those ideals T : N---->Omega such that for all n
Tn = Inf{Tm | m > n}.
The precise meaning of this definition is explained in the given reference, as
well as in Bill's messages. I hope that this is correct.

                   Aurelio Carboni