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
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.
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.
participants (2)
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carboni＠vmimat.mat.unimi.it -
Steve Vickers