[Logica-l] Filter-powers

[Eduardo Ochs]

Oi Hugo!

On 8/2/06, Hugo Luiz Mariano <hugomar em ime.usp.br> wrote:
> Oi Eduardo,
>
> Muita coincidência!!! O Odilon e eu acabamos (há 10 minutos) de falar sobre
> esse artigo!!! E´ uma artigo do Anders Kock e do Mikelsen de 1985 (+-) cujo
> titulo é, se nao me engano ``Non-standard fatorizations in Topos´´.
> Coincidendemente tambem falavamos da necessidade, até premência, de uma maior e
> mais frequente interacao entre os interessadas em categorias e lógica no
> Brasil. Vou perguntar pro Odilon se ele se lembra aonde saiu o artigo.
> Vamos continuar conversando!
>
> Abraco,
> Hugo

Saiu numa SLNM... o Joao Marcos me mandou as reviews de todos os
artigos do Anders Kock que ele encontrou...

MR0480685 (58 841) 18C99 02H20
Kock, Anders ; Mikkelsen, Chr. Juul
Topos-theoretic factorization of non-standard extensions.
Victoria Symposium on Nonstandard Analysis (Univ. Victoria,
Victoria, B.C., 1972), pp. 122-143. Lecture Notes in Math., Vol. 369,
Springer, Berlin, 1974.
That Robinson's nonstandard analysis is one of the various develop-
ments of mathematics which are capable of being interpreted within
topos theory was recognized readily by F. W. Lawvere. An enlarge-
ment of a superstructure results in a change of logic, hence it should
be best viewed as a change of topos. But the main aspect of non-
standard analysis is, as the authors put it, "the contradiction that
higher order properties are preserved yet not preserved by nonstan-
dard extensions". It is this basic principle that they set to analyse
in the form of a factorization theorem for first-order logic preserving
functors between elementary toposes. Their main theorem says that
any such functor : E  E0 admits a factorization E  E  E0 where
the first functor preserves higher-order logic while the second functor
preserves elements (as well as first-order logic). This type of theorem
seems to be comparable to Theorem 4.1 in the paper by A. Robin-
son and E. Zakon [Applications of model theory to algebra, analysis
and probability (Internat. Sympos., Pasadena, Calif., 1967), pp. 109-
122, Holt, Rinehart and Winston, New York, 1969; MR0239965 (39
#1319)] rather than being akin to the main theorem of nonstandard
analysis, which says that every superstructure has enlargements of a
certain kind. For a proof in categorical terms of the latter, see an arti-
cle of G. Reyes [Studies in algebraic logic, pp. 143-204, Math. Assoc.
Amer., Washington, D.C., 1974; MR0360735 (50 #13182)]. The pre-
sent paper is rather an analysis of a given situation, embodied in the
functor : E  E0 . If  is the endofunctor Iof the category S of sets,
which assigns to a set X the ultrapower X/D, where D is a non-
prIncipal ultrafilter on a set I , then E is equivalent to the category
S/D. The abstract definition of E , however, little resembles this
construction. A first approximation which is also in line with Robin-
son's theory is to take the category whose ob jects are of the form X
for an ob ject X of E, and whose morphisms are internal maps. The
definition of "internal" and "standard" entities is the topos version of
similar notions introduced by Robinson, and establishing their prop-
erties is one of the main aspects of this paper. However, the resulting
category fails to be a topos. The next approximation is to take the
categorical version of the pseudo-ob jects and pseudomorphisms of M.
Machover and J. Hirschfeld [Lectures in non-standard analysis, Lec-
ture Notes in Math., Vol. 94, Springer, Berlin, 1969; MR0249285 (40
#2531)]. This is the required E , and the main part of the proof of
the main theorem is to prove that E is a topos. Since the contents of
this paper were available earlier in preprint form [the authors, "Non-
standard extensions in the theory of toposes", Aarhus Univ. Preprint
Series No. 25, Aarhus Univ., Aarhus, 1971/72], the authors here tend
to omit some of the proofs or else replace them by heuristic argu-
ments; this makes the paper highly readable, and it seems particularly
suited to an audience of logicians. It is a charming paper on a charm-
ing sub ject, not much explored further elsewhere in the context of
topos theory, except in the above-mentioned paper of G. Reyes and, in
a great deal more generality, in a paper of H. Volger [Model theory and
topoi, pp. 87-100, Lecture Notes in Math., Vol. 445, Springer, Berlin,
1975; MR0376809 (51 #12984)]. The ultraproduct construction itself
was, of course, described internally in one of the early works in topos
theory, by its creators, F. W. Lawvere and M. Tierney [Categories
and commutative algeba (C.I.M.E., III Ciclo, Varenna, 1971), pp. 249-
326, Edizioni Cremonese, Rome, 1973; MR0354800 (50 #7277)]. But
the relationship between their construction and the one in the pa-
per under review does not seem to have yet been made explicit. The
paper also contains a heuristic proof, discovered by the first author,
that exponentiation can be defined in terms of power ob ject forma-
tion and so one need only assume the latter plus finite inverse limits
and the existence of a subob ject classifier in order to define the no-
tion of elementary topos. The full proof occurs in the earlier version
of the paper [op.cit.]. From the point of view of logical applications,
as is the case of the present paper, it is certainly more suitable to
work with power ob jects; this may also be a reason why it occurred
in this context.
  {For the entire collection see MR0472459 (57 #12159).}
                                   Marta C. Bunge (Montreal, Que.)

Acho que as SLNMs nao estao online, entao vou ter que
abrir as caixas de artigos que eu deixei na casa dos meus pais
e procurar a minha copia...

Nem sei se vou poder mexer com esse artigo agora - tou concentrado
nesse aqui - http://www.dcs.ed.ac.uk/home/mxh/csl94.ps.gz - e estou
traduzindo-o pra linguagem de diagramas que eu desenvolvi na
minha tese... mas, bom, sei la', vamos ver. :-)

  [], Edrx

P.S.: qual e' o e-mail do Odilon? Eu fiquei de mandar umas coisas -
musicas - pra ele desde o EBL e ainda nao mandei...