From catdist@mta.ca Ukn Jan 3 15:36:39 1994
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From: categories
Subject: (not entirely) routine distribution
To: categories
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***************************************************************************
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* Change in Originating Machine for categories list. *
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With this posting the originating machine for categories changes (to
nimble.mta.ca [138.73.1.253].) It is hoped that the change will be
seamless, but please bear with us if there are some shortterm glitches.
There is NO CHANGE in address for the list, nor in the
categoriesrequest@mta.ca alias which should now be working. The only
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 the `To:' field will now read `categories ' (and not
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You may also note some other changes in the mailheader as you receive it.
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will be posted unless the moderator clearly understands a different
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Please inform me of any problems you experience. The usual routine
distribution follows. ( ... and Happy New Year to all.)
Bob Rosebrugh
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
This is the routine distribution for the categories mailing list. It is the
file routine.dist in pub/categories at sun1.mta.ca and was last updated on
January 2, 1994.
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Bob Rosebrugh Phone: +15063642538
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From catdist@mta.ca Ukn Jan 3 22:18:43 1994
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From: categories
Subject: doctoral scholarship
To: categories
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Date: Tue, 4 Jan 1994 12:45:53 +1100
From: Barry Jay
Doctoral Scholarship
School of Computing Sciences
University of Technology, Sydney
**************************************************
*** Funding is now available for three years ***
**************************************************
The scholarship is linked to the "Shape" project, whose goal is to
design and implement an efficient, typed programming language for
vectors, matrices, and other data structures, based on the separation
of data from shape. The project combines both the development of the
theory of shapely types with their incorporation in a programming
language.
The project is based at UTS, but has participants in Calgary and
Edinburgh.
Applicants will have completed an undergraduate degree at the Honours
level. They should also have experience in programming language
implementation or semantics, or in category theory.
The stipend is $18,679. Funding is for three years, subject to review
after the first year. Applicants who are Australian citizens, or
permanent residents who have lived continuously in Australia for the
last twelve months, would not be required to pay student fees, or
taxes on the stipend.
For further information, contact Barry Jay (cbj@socs.uts.edu.au).
Applications should contain a curriculum vitae and the names,
addresses, phone numbers and email of two or three referees from whom
confidential reports can be obtained. They should be sent to:
Dr C. Barry Jay
School of Computing Sciences
University of Technology, Sydney
PO Box 123
Broadway
Australia
2007
Ph: (02) 3301814
for receipt by 21st January, 1994.
From CATEGORIES@mta.ca Ukn Jan 11 19:19:43 1994
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Date: Tue, 11 Jan 1994 19:26:14 0400 (AST)
From: CATEGORIES@mta.ca
To: rrosebrugh@macc2.mta.ca
MessageId: <940111192615.2040f68c@mta.ca>
Subject: strict epi(morphic) family
Status: RO
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Date: Tue, 11 Jan 94 13:41:41 +0100
From: Thomas Streicher
Does anybody know what is meant by "strict epi" or "strict epimorphic
family" ?
Maybe it's standard but I couldn't find it in any book available to
me !
Thomas Streicher
From CATEGORIES@mta.ca Ukn Jan 11 20:56:17 1994
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From: CATEGORIES@mta.ca
To: rrosebrugh@macc2.mta.ca
MessageId: <940111194951.2040fea4@mta.ca>
Subject: Letter re: The Death of Proof
Status: RO
XStatus:
Note from moderator: Jim sent me a copy the following letter in December, and I
received it only last week, with a note mentioning that colleagues suggested it
be circulated. The subject is still current despite the further delay awaiting
retyping of the letter.
Bob Rosebrugh.
++++++++++++++++++++++
Date: Tue, 11 Jan 1994 9:48:00 0400 (AST)
From: Jim Lambek
October 18, 1993
The Editor
Scientific American, Inc.
415 Madison Avenue
New York, NY 10017
USA
Dear Sir:
As much as I enjoyed reading ``The death of proof'' by John Hogan [October
1993], I feel I should take exception to the statement attributed to Thurston
that Godel's incompleteness theorem implies that ``it is impossible to
codify mathematics.'' This may be the view accepted by most mathematicians,
and perhaps by Godel himself, but it depends on the assumption, implicit
in Godel's argument, that the universe of mathematics (by this I mean
whatever is described by the language of mathematics) obeys the following
principle: if a formula A(x) becomes true when x is replaced by a
numeral S ... S0 (successor of ... successor of zero), then the universal
statement `for all numbers x, A(x)' is true. This principle is indeed
accepted by most mathematicians, but not by intuitionists, followers of
L.E.J. Brouwer, who believe in the truth of the universal statement only if
the special instances can be seen to be true in a way that is uniform as
regards the number of times the successor symbol S appears. How can such
an intuitively evident principle fail to hold? Well, intuitionists do accept
the following modified principle: if the existential statement `for some
number x, B(x)' is true, then some special instance B(S ... S0) must
be true. This looks very much like the principle they rejected, but the two
principles are only equivalent according to the rules of classical, not
intuitionistic, logic.
So, what is the universe of mathematics? If we take the language of
mathematics to be some form of intuitionistic type theory, we would hope
that, among the models
of this language, there is a preferred one, call it the
universe of mathematics. Indeed, such a model can be constructed from the
language itself, and it satisfies the above principle in its modified form.
That it does not satisfy the original principle is, in fact, what Godel
proved. For classical type theory, no one has yet constructed a preferred
model, or any model for that matter, although many such models can be shown
to exist in the classical, but not intuitionistic, sense of ``existence.''
Jim Lambek
Montreal, Quebec
From CATEGORIES@mta.ca Ukn Jan 11 22:38:16 1994
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Date: Tue, 11 Jan 1994 19:20:09 0400 (AST)
From: CATEGORIES@mta.ca
To: rrosebrugh@macc2.mta.ca
MessageId: <940111192009.2040f573@mta.ca>
Subject: Sydney Bushfires
Status: RO
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Date: Tue, 11 Jan 94 14:22:20 +1100
From: street@macadam.mpce.mq.edu.au
Dear CATEGORIES
Many of our kind colleagues have been concerned for the safety
of Sydney category theorists. I have had many enquiries by email
and phone.
We are all safe and without loss of personal property, to the best
of my knowledge.
The fires have been the worst in New South Wales post european
settlement, as far as the area covered and number of fire fronts
at one time. Because of the remarkable efforts of the fire fighters
(many voluntary, from all over Australia & even some from New Zealand),
the death toll was kept down to around 5, including 2 fire fighters,
and there were 180 houses lost. Of course, there were many injuries
and other property losses. The Royal National Park south of Sydney
was 95% burnt; much will begin to regenerate with the next rain, but
many animals are dead and the rainforest regions will take 250 years
to return to the state they were in last week. The fire is still not
dead.
Last Thursday during lunch with Todd Trimble (our new Research Fellow
from Rutgers) at the Macquarie University Union we saw the beginning
of the fire in Lane Cove National Park. This was responsible for
the loss of many homes around South Turramurra, West Pymble, West
Killara and West Pymble. During lunch with Todd on the Friday, helicopters
began taking water from a small lake near the Union building to
dump on the fire. Macquarie Centre (a nearby shopping mall) was
evacuated on Friday evening as was the Graduate School of Management;
but these were all saved. The Kuringgai Campus of the University of
Technology, Sydney, in Lindfield had considerable fire damage (my
wife, Margery, has done various courses there). Our Head of
School, the number theorist Alf van der Poorten, was one small block
away from destroyed houses and bushland. This fire is essentially dead now.
The fire in the Blue Mountains west of Sydney rages on out of control,
but the efforts to save settlements by backburning and fire fighting have
mostly succeeded.
The fire in Kuringgai National Park north of Sydney has charged back
and forth with the wind. It destroyed houses on the Pittwater
Penninsula on Saturday during the westerly winds then headed back
south west with the change of wind on Sunday. Margery was working
Sunday afternoon at Belrose Library; as soon as she arrived (by a
nonstandard route because of road closures just north of Max Kelly's
house on Mona Vale Road), Belrose hit the news as the centre of
activity. Luckily it stayed north of the Library. But the northern
parts of St Ives (Max's suburb), Turramurra (my suburb) and Wahroonga
(Bob Walter's suburb  Bob is in Milan, however) were warned to
prepare for evacuation. A large nursing home was evacuated. Cooler
conditions saved the day. The fire is still active.
Yesterday there was some rain which hampered backburning efforts a
bit; it was not enough to put out fires, but the overcast conditions
and increased humidity helped. Today is still fairly cool (31 degrees
C predicted maximum for Sydney) and more humid.
Throughout it all, life goes on. In fact, life began. Another beautiful
daughter (9lb 12oz), Florence, arrived 'midst the flames on Friday for
Dominic and Sally Verity.
Other personal news: Bart Jacobs is visiting Macquarie. His wife, Joke,
was here, but left last Friday returning to Utrecht. Awaiting her at
Amsterdam airport were reporters seeking news of the fires. Joke was
on Netherlands National TV. Also, Mike Johnson's student, Richard
Buckland is building a house in a little part of the Hawkesbury
River (north of Sydney) with only access by boat. Foundations, and
floorboards were in place. Last weekend he was going to put up
walls, and a lot of wood was at the site. He made several attempts
to visit the site over the weekend but was turned back. The good
news is that everything is untouched by fire, although it had gone
close.
Today (Tuesday) is now quite hazy and smokey. Hotter weather is
predicted. We are told not to become complacent.
Finally, I thought you might find of interest the following letter to
the newspaper written by my (22 years old tomorrow) son, Arthur:

\documentstyle[12pt,a4wide]{letter}
\begin{document}
\signature{Arthur Street\\32 Katina St\\Turramurra 2074}
\begin{letter}{
Letters Editor\\
GPO Box 3771\\
Sydney 2001}
\opening{Dear Editor,}
In the wake of the bushfires I was intrigued to see the media performing its
usual postcrisis contortions in an attempt to find people to blame for
the destruction.
I can't help but feel that no matter how much we backburn
and no matter how draconian the penalties for flicking cigarette butts out
car windows, bushfires are inevitable and always have been.
As was mentioned by the {\em Herald} recently, Captain Cook was
flabbergasted by the number of fires he saw raging up and down the east
coast of Australia.
Therefore I propose that, rather than blaming individuals and dare I say
the greenies, we should blame our forebears for imposing on this
`continent of fire' a foreign culture not suited to the conditions.
Instead of concreting over the last of our urban bushland, which no doubt
some developmentmotivated doommongers will be advocating,
the housing developments at the fringes of Sydney should be built to
withstand the worst of our bushfires. Perhaps they could be built largely
underground, which would also provide a cheap form of temperature control.
Who knows, if it were to catch on, the resulting unique
architecture could help to forge the Australian identity of the 21st
century.
\closing{Yours sincerely,}
\end{letter}
\end{document}

Best regards,
Ross
From CATEGORIES@mta.ca Ukn Jan 12 23:52:14 1994
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Date: Wed, 12 Jan 1994 23:54:32 0400 (AST)
From: CATEGORIES@mta.ca
To: rrosebrugh@macc2.mta.ca
MessageId: <940112235432.20410a57@mta.ca>
Subject: question on homotopies
Status: RO
XStatus:
Date: Wed, 12 Jan 94 13:16:22 EST
From: Michael Barr
In the category of chain complexes (over some abelian category)
it is clear that if you identify homotopic arrows, you will also
invert homotopy equivalences. Does anyone know if the converse is
true? If you invert homotopy equivalences, do you wind up identifying
homotopic arrows? What if you replace homotopy by homology? Does
inverting maps that induce isomorphism on homology have the effect of
identifying maps that induce the same map on homology?
Michael
From CATEGORIES@mta.ca Ukn Jan 12 23:52:22 1994
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Date: Wed, 12 Jan 1994 23:55:02 0400 (AST)
From: CATEGORIES@mta.ca
To: CATEGORIES@mta.ca
Cc: pratt@cs.stanford.edu
MessageId: <940112235502.20410c64@mta.ca>
Subject: An object of formulas
Status: RO
XStatus:
Date: Wed, 12 Jan 94 19:38:12 0500
From: "Charles F. Wells"
This message concerns the idea that one could carry out the
usual construction in classical logic of formulas, terms,
theories, interpretations, and so on in a suitable category that
has recursion. Formulas, terms and proofs are all inductively
defined, after all. For example, one might have an object of
formulas that is the solution of a recursion problem and an
arrow "interpretation" (also the solution of a recursion
equation) from that object to a category object.
I believe I recently saw a paper that does something like this,
but I can't remember it and can't find it. Although I saw it
recently, it may not be a recent paper; I believe I came across
it in the process of accumulating papers for my outline of
sketches.
Anyone know of any work like this?

Charles Wells, Department of Mathematics, Case Western Reserve University
10900 Euclid Avenue, Cleveland OH 441067058, USA
Phone 216 368 2880 or 216 774 1926
FAX 216 368 5163
From CATEGORIES@mta.ca Ukn Jan 13 02:31:22 1994
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Date: Wed, 12 Jan 1994 23:54:31 0400 (AST)
From: CATEGORIES@mta.ca
To: rrosebrugh@macc2.mta.ca
MessageId: <940112235432.20410c59@mta.ca>
Subject: re: strict epi(morphic) family(3 responses)
Status: RO
XStatus:
Date: Tue, 11 Jan 94 18:11:05 EST
From: Walter
Strictly epic families (of morphisms with common codomain) are used in
SGA4. I used them again in my paper with Reinhard Boerger in the Cahiers
(vol 32, 1991, pp257ff) which contains exact definitions and a bit of
discussion. In the case of singleton families, these are regular epis
in the sense of GabrielUlmer and of Kelly.
I hope that this answers Thomas Streicher's question.
Walter Tholen.
++++++++++++++++++++++++++++++++++++++++++++
Date: Tue, 11 Jan 94 17:07:51 EST
From: Michael Barr
It is, to my knowledge standard and is identical to extremal epi or
extremal epimorphic family. For one, it means it factors through no
proper subobject of the codomain. For a family, all with the same
codomain, it means there is no proper subobject of the codomain
that factors all of them. In a regular category, a strict epi is
regular. In a wide complete category (every class of subobjects has
an intersection), a strict epi is in a certain sense a composite
of a class of regular epis. You may need cokernel pairs to do that.
I don't know any such simple characterization for families.
Michael
+++++++++++++++++++++++++++++++++++++++++++
Date: Wed, 12 Jan 94 09:52:25 +0100
From: Markus Wolf
> Date: Tue, 11 Jan 94 13:41:41 +0100
> From: Thomas Streicher
> Does anybody know what is meant by "strict epi" or "strict epimorphic
> family" ?
> Maybe it's standard but I couldn't find it in any book available to
> me !
> Thomas Streicher
Hmm, not quite sure, but in the book "Category Theory, An Introduction" by
H. Herrlich and G. Strecker there is a definition of strict mono in the
exercises.
According to this text a strict monomorphism is a morphism $f$ such that:
whenever $h$ is a morphism with the property that for all morphisms
$r$ and $s$, $r\circ f= s\circ f$ implies that $r\circ h=s\circ h$, then
there exists a unique morphism $k$ such that $h=f\circ k$. Probably you get
a strict epi if you modify the condition according to the property of
epimorphisms being righcancellable instead of leftcancellable.
There was no reference to "strict monomorphic family" :(
Markus Wolf
From CATEGORIES@mta.ca Ukn Jan 13 03:38:59 1994
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Date: Wed, 12 Jan 1994 23:54:32 0400 (AST)
From: CATEGORIES@mta.ca
To: rrosebrugh@macc2.mta.ca
MessageId: <940112235432.20410e5a@mta.ca>
Subject: Re: Letter re: The Death of Proof
Status: RO
XStatus:
Date: Tue, 11 Jan 94 22:13:52 PST
From: pratt@cs.stanford.edu
I am very much in favor of constructive logic in the sense of proof in
lieu of truth. This replaces truthvalued entailments, that is,
implication as a preordering of propositions, with setvalued
entailments, that is, implication as homsets whose elements are
interpreted as proofs, or moves in a space of perspectives. This
branch of logic is a rich application area for category theory, whose
cartesian closed fragment is nicely developed in Jim and Phil's book.
To the same degree, I am opposed to the mysticism implicit in the
traditional "it may sound the same but it's provably different" defense
of propositional intuitionism. Goedel defused this argument along
lines that can be made quite pragmatic as follows.
Intuitionistic arithmetic is just classical arithmetic with some
"surely"'s strategically placed in the wording of some theorems, where
"surely" abbreviates "not not". Realworld customers of mathematics
pay no attention to these "surely"'s, for the simple reason that, while
they know full well what to do with the theorems, they have no idea to
what use to put the "surely"'s. No engineering project has ever been
made one whit more hazardous by casual neglect of "surely"'s appearing
in its intuitionistically correct arithmetic. Deriving those theorems
classically yields no more than what can be obtained by treating all
the "surely"'s in the intuitionistic derivations as noise words to be
ignored.
From the point of view of correctness then, "surely" is merely harmless.
From a computational point of view a marginally more concrete statement
is possible. Dropping "surely" from pure propositional logic reduces
the computational complexity of its satisfiability problem from
PSPACEcomplete (Statman) to NPcomplete (Cook). Thus "surely," and
hence intuitionism in the permittedmiddle sense, does nothing to
render the propositional content of an argument any easier to follow,
and may well make it harder.
If Jim has something more constructive in mind than this, e.g. along
the lines of categorial logic, it would be nice to understand the
details. In a suitable setting for example, it may be possible to
imbue "surely" with enough realworld significance to change the
practical import of arithmetic statements using it.
I maintain that any defense of this brand of intuitionism can be
clarified, and hence made more convincing, by phrasing it as a
raisond'etre for "surely," or double negation, which, as Goedel was
the first to see, conveniently localizes the significance of
intuitionism by presenting it as an expansion of the language of
classical logic with a unary operator.

Vaughan Pratt
 End of Forwarded Message
From CATEGORIES@mta.ca Ukn Jan 13 21:30:15 1994
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Date: Thu, 13 Jan 1994 21:29:55 0400 (AST)
From: CATEGORIES@mta.ca
To: rrosebrugh@macc2.mta.ca
MessageId: <940113212955.20411c5e@mta.ca>
Subject: Re: An object of formulas
Status: RO
XStatus:
Date: Thu, 13 Jan 1994 10:02:22 0500 (EST)
From: MTHFWL@ubvms.cc.buffalo.edu
Perhaps what Charles Wells was reading was SLNM 661 on "indexed categories",
where recursion is put to such uses.
From CATEGORIES@mta.ca Ukn Jan 13 23:14:07 1994
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Date: Thu, 13 Jan 1994 21:28:52 0400 (AST)
From: CATEGORIES@mta.ca
To: rrosebrugh@macc2.mta.ca
MessageId: <940113212853.2040f5aa@mta.ca>
Subject: Re: question on homotopies
Status: RO
XStatus:
>From MAS013@bangor.ac.uk Thu Jan 13 13:16:59 1994
Date: Thu, 13 Jan 94 9:03 GMT
>From Tim Porter.
In reply to Mike Barr's question, think of a cylinder generating the
homotopy. The inclusions into the two ends are homotopy equivalences.
(There are various cylinders possible; one due to Kleisi some years
back.) Invert the homotopy equivalences and homotopic maps become
identified. This was discussed in detail and more generality in
Grothendieck's Pursuing Stacks.
Happy New Year to all and sundry,
Tim
mas013@uk.ac.bangor
From CATEGORIES@mta.ca Ukn Jan 14 02:57:36 1994
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Date: Thu, 13 Jan 1994 21:28:02 0400 (AST)
From: CATEGORIES@mta.ca
To: rrosebrugh@macc2.mta.ca
MessageId: <940113212803.20411ada@mta.ca>
Subject: Re: strict epi(morphic) family, question on homotopies
Status: RO
XStatus:
Date: Thu, 13 Jan 94 14:28:52 EST
From: Vladimir Voevodsky
Does anybody know what is meant by "strict epi" or "strict epimorphic
family" ?
Maybe it's standard but I couldn't find it in any book available to
me !
Thomas Streicher
For a category which has fiber products the definition looks as follows
(there are LATEX notations below).
A family of morphisms $\{f_i:X_i\rightarrow Y\}$ in a category $C$ is
called a stric epimorphic family if for any object $Z$ in $C$ the
set $Hom(X,Z)$ is the equalizer of the maps $\prod_i
Hom(X_i,Z)\rightarrow \prod_{i,j} Hom(X_i\times_Y X_j,Z)$ induced by the
projections.
If $C$ does not have fiber products one has in the definition above to
replace $X_i\times_Y X_j$ by the fiber product of the corresponding
representable functors and to consider morphisms in the category of
functors from $C^{op}$ to $Sets$ (taking $Z$ to be a representable
functor).
A more sophisticated way to say the same thing is to say that a strict
epimorphic family is a covering family in the canonical topology on $C$.
All this should be in SGA4 and the definition for categories with fiber
products should be in SGA1.
In the category of chain complexes (over some abelian category)
it is clear that if you identify homotopic arrows, you will also
invert homotopy equivalences. Does anyone know if the converse is
true? If you invert homotopy equivalences, do you wind up identifying
homotopic arrows? What if you replace homotopy by homology? Does
inverting maps that induce isomorphism on homology have the effect of
identifying maps that induce the same map on homology?
Michael
The answer to the first question I guess is positive (and we do not
need the original category to be abelian  just additive). The proof should
look as follows. Let $f:X\rightarrow Y$ be a morphism which is homotopic
to zero. We have to show that it is zero in the category localized with
respect to homotopy equivalences. It follows from the fact that it can
be factorized through the cone of the identity morphism $X\rightarrow X$
which is homotopy equivalent to zero.
The answer to the second one is most definitely negative. In fact the
statement is true if and only if the original abelian category is
semisimple. The localization with respect to "homological equivalences"
i.e. quasiisomorphisms is the derived category. Any extension in the
original abelian category gives us a morphism in the derived one which
is zero on (co)homology.
Vlaidmir Voevodsky.
From catdist Fri Jan 14 12:52:05 1994
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Date: Fri, 14 Jan 1994 12:51:20 +0400 (GMT+4:00)
From: categories
Subject: Fuzzy +
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Date: Fri, 14 Jan 1994 09:05:25 +0100
From: Axel Poigne
I yesterday attended a talk about fuzzy logic (I know this to be
``degoutant'', but...) where a ``Lukaciewicz norm'' was discussed as a
tnorm.
If I recollect correctly, a tnorm is a has a binary operator _\wedge_
which is associative, commutative, and monotonic, the latter being a
mystery, the order being due to the real interval [0,1]. Moreover it seems
that a negation \neg is assumed to exist, since an operator \vee is
defined by de Morgan law. Quite clearly, min determines a norm as well as
the multiplication. It seems to be an assumption that negation is always
\neg a = 1a.
Now the Lukaciewicz norm is of the form a \wedge b = min{a + b, 1}. As
consequence, a \vee b = max{a + b  1, 0}. This norm satisfies a \wegde
\neg a = 0 and a \vee \neg a = 1, but \vee and \wedge are not
distributive, which is true for the other norms. (I hope this to be a
correct recollection of what I heard)
Trying to make head and tail of this, I wonder whether one really should
say that one has a lower semilattice for the order, or even an Heyting
algebra (in fact the \sqcap and \bigsqcup is about in all the arguments),
and just add a binary monotonic, etc operator _ \otimes _ (replacing the
\wedge in the tnorm). This structure rather looks like a quantale (units
are available as well). I have no idea how negation fits the picture, but
it reminds me of classical linear logic.
Does this ring a bell ? I am just puzzled, having no idea about fuzzy
logic, and little knowlege about linear logic. I know that Michael Barr
has written a paper on Fuzzy sets as toposes but he uses only geometric
logic, meaning a Heyting algebra.
Axel
A related question : these people seem to use \bigsqcup in general to
compute suprema. It appears to be more consistent to use \bigoplus on
occasions. How would this be defined in linear logic ? (Sorry, my linear
logic is very poor)
From catdist@mta.ca Ukn Jan 14 13:18:16 1994
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From: categories
Subject: I like my coffee crisp
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Date: Fri, 14 Jan 94 9:21:59 EST
From: Al Vilcius
This is an appeal for help to my friends and acquaintances on
CATEGORIES:
I have been enlisted to give a talk on "Fuzzy vs. Probability
Theory" to an audience in finance whose backgrounds include
applied mathematics, physics, engineering, and finance. The
audience is not familiar with categories, no less toposes, which
means that subtleties such as adding fuzzy equality to yield
variable sets (sheaves) would be lost. Nevertheless, they are
intrigued by the "fuzzy stuff" that is currently popular.
My predicament is then to choose between:
(1) torturing my conscience by giving an insubstantial and
superficial talk on memberships vs distribution functions;
(2) torturing the underlying mathematics into layman's prose.
I am hoping that some of the learned readers of CATEGORIES may
have already performed torture # 2 in a humane fashion (either in
public or in private) and have some material and/or suggestions
on how best to commit this heinous act.
My preferred approach would be a la M. Barr via variable sets and
sheaves, combined with the description of fuzzy and probabilistic
algebraic theories given by E. Manes. I am already aware of many
other fine (and some not so fine) works on fuzzy sets and fuzzy
logic, however, I don't know how to make these understandable to
noncategorists. I may well have to resort to torture # 1, but
would like to avoid doing so if possible.
All comments and suggestions, either privately to me at
vilcius@clid.yorku.ca or publicly on CATEGORIES, on how I could
have my "coffee crisp" would be most welcome.
Thank you ............................... Al Vilcius, Toronto
/\ /
/ \ /
/>\ /
/ \/
From catdist@mta.ca Ukn Jan 14 22:00:51 1994
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Date: Fri, 14 Jan 1994 21:53:59 +0400 (GMT+4:00)
From: categories
Subject: simple characterization of weak cartesian closedness
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Date: Fri, 14 Jan 94 18:55:12 +0100
From: Thomas Streicher
I wonder whether the following trivial observation is generally known
:
a category C with finite products is WEAKLY CARTESIAN CLOSED iff
for all objects A , B in C the functor C( _ x A , B) is a
retract of a representable functor.
(The embedding part of the retraction gives a choice of functional
abstraction which is stable under substitution) and the projection
part gives evaluation).
Especially this entails that if C has splitting of idempotents then
the notions of cartesian closedness and weakly cartesian closedness
are equivalent.
I don't think that the remark above is a deep insight !!
BUT usually people refer to the quite heavy machinery of Hayashi's
semifunctors when they speak about the categorical semantics of typed
lambda calculus without etarule.
Thomas Streicher
From catdist@mta.ca Ukn Jan 14 22:13:22 1994
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Date: Fri, 14 Jan 1994 22:04:18 +0400 (GMT+4:00)
From: categories
Subject: cantorbernstein
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Date: Fri, 14 Jan 94 15:44:32 EST
From: Peter Freyd
Pierre Ageron asked (way back on 1 Dec):
The statement and the ACfree proof of CantorSchroederBernstein's theorem
have obviously some categorical content. Has this been already investigated ?
The property, when it holds, is an important property on the category. It is,
however, a rare property.
There are, as usual, different ways to interpret the property in general
categories. I would opt for the following:
CANTORSCHROEDERBERNSTEIN PROPERTY: If two objects be retracts of each other
they are necessarily isomorphic.
I trust that CSB holds for any boolean topos. (Anyone want to confirm?)
Kaplansky in his booklet on infinite abelian groups pointed out that CSB holds
in the category of countable torsion abelian groups (as a consequence of the
Ulm invariants). He raised it as one of three test problems for advances in
the theory of abelian groups. Does CSB continue to hold, for example, if
countablility is dropped? (Kaplansky did not, of course, talk about retracts.
He talked about two groups appearing as direct summands of each other.)
Someone found a counterexample in the latter 50's. (Anybody know who?)
If _A_ and _B_ are categories, _A_ a retract of _B_, it is routine that
a counterexample for CSB in _A_ is transported to a countexample in _B_.
_Abelian_Groups_ is a retract of _Topological_Spaces_ (via Moore spaces and
homology) hence there are a pair of spaces which appear as retracts of each
other but are different enough to have different homology groups. That fact
became better known in the late 50's than the fact about abelian groups. (And
in the late 50's it was damned difficult to explain why it should be viewed
as a trivial corollary.)
There's a stronger property: if two objects be retracts of each other the
retraction maps are isomorphisms.
The two most immediate examples are the categories of finite sets and of
finite dimensional vectors spaces. But note that any category that is
locally finite (i.e. all homsets are finite)or any linear category that
is locally finite dimensionalimmediately inherits the property. By moving
to a 2category setting one may state the obvious general theorem of
which these are special cases.
I am not sure if any of this should be viewed as having "categorical content."
best thoughts,
peter
From catdist@mta.ca Ukn Jan 14 22:19:20 1994
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From: categories
Subject: syntactic criterion for join?
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Date: Fri, 14 Jan 1994 16:47:01 0500
From: David Espinosa
Does anyone know a syntactic criterion for the existence of a natural
transformation join : TTA > TA for a given endofunction T built from
+, *, > ?
There is a wellknown (correct me if I'm wrong) syntactic criterion
for covariance which determines whether T can be extended to an
endofunctor. Can this criterion be extended to the existence of join?
Also, does anyone know a reference for the covariance criterion?
David
From catdist@mta.ca Ukn Jan 14 22:23:22 1994
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Date: Fri, 14 Jan 1994 22:15:51 +0400 (GMT+4:00)
From: categories
Subject: Re: Fuzzy +
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Date: Fri, 14 Jan 94 13:18:44 EST
From: Michael Barr
One thing to say is that I did not write a paper on fuzzy sets as toposes,
but someone named Eytan did. I wrote a paper on fuzzy sets as nontoposes
and it differs from Eytan's in being correct. On the other hand, fuzzy
sets are a quasi topos, which means they do have a first order logic.
That said, it has to admitted that the first order logic is probably not
what they really had in mind as fuzzy logic and what they did have in
mind (using operators like truncated sum and negations like  minus is
closer to linear logic than to classical, even intuitionistic classical,
logic. I once started to write a paper on this, but have not completed
it it; maybe one day I will. And, BTW, Andy Pitts, unbeknownst to me,
also once wrote a paper on fuzzy sets as a nontopos. His is also correct.
Michael
From catdist@mta.ca Ukn Jan 14 22:27:53 1994
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From: categories
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Date: Fri, 14 Jan 94 12:28:42 PST
From: "Michael J. Healy (206) 8653123"
> Date: Fri, 14 Jan 94 9:21:59 EST
> From: Al Vilcius
>
> This is an appeal for help to my friends and acquaintances on
> CATEGORIES:
>
> I have been enlisted to give a talk on "Fuzzy vs. Probability
> Theory" to an audience in finance whose backgrounds include
> applied mathematics, physics, engineering, and finance. The
> audience is not familiar with categories, no less toposes, which
> means that subtleties such as adding fuzzy equality to yield
> variable sets (sheaves) would be lost. Nevertheless, they are
> intrigued by the "fuzzy stuff" that is currently popular.
>
> My predicament is then to choose between:
>
> (1) torturing my conscience by giving an insubstantial and
> superficial talk on memberships vs distribution functions;
>
> (2) torturing the underlying mathematics into layman's prose.
>
> I am hoping that some of the learned readers of CATEGORIES may
> have already performed torture # 2 in a humane fashion (either in
> public or in private) and have some material and/or suggestions
> on how best to commit this heinous act.
>
> My preferred approach would be a la M. Barr via variable sets and
> sheaves, combined with the description of fuzzy and probabilistic
> algebraic theories given by E. Manes. I am already aware of many
> other fine (and some not so fine) works on fuzzy sets and fuzzy
> logic, however, I don't know how to make these understandable to
> noncategorists. I may well have to resort to torture # 1, but
> would like to avoid doing so if possible.
>
> All comments and suggestions, either privately to me at
> vilcius@clid.yorku.ca or publicly on CATEGORIES, on how I could
> have my "coffee crisp" would be most welcome.
>
> Thank you ............................... Al Vilcius, Toronto
>
> /\ /
> / \ /
> />\ /
> / \/
>
>
I have a related predicament. I'm an industrial mathematician with a
need to learn what I can as soon as possible about a mathematical
background for fuzzy logic. I am also furiously learning what I can
about category theory and logic in connection with some work in formal
methods for software engineering and machine learning (neural networks).
So I really need to find an appropriate, nononsense (i.e., mathematical)
formalism that meets all these requirements; given that, I can afford to
invest considerable effort learning it. My current choice is to study
categorical or categoryrelated theories, and am currently reading up
on Steven Vickers' work on topological systems as well as Goguen and
Burstalls' work on institutions. If anybody has information that might
help, or could elaborate on your reply to Al Vilcius so that a categorical
novice might understand as well, I would be most grateful. I did study
topology and algebra in grad school many years ago.
Thank you,
Mike Healy
mjhealy@atc.boeing.com
From catdist@mta.ca Ukn Jan 16 12:52:30 1994
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Date: Sun, 16 Jan 1994 12:37:25 +0400 (GMT+4:00)
From: categories
Subject: Acyclic models
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Date: Sat, 15 Jan 94 11:43:33 EST
From: Michael Barr
There have been two apparently quite different categorical versions of
acyclic models. The first, as found for example in BarrBeck, COCA,
1966, says that if K = {Kn}, augmented over K(1) and similarly L were
chain complex functors and G is a cotriple such that (Kn)epsilon: (Kn)G
> Kn has a natural splitting when n >= 0 and if the complex LG >
L(1)G > 0 has a natural contracting homotopy, then any natural
transformation K(1) > L(1) can be extended to a unique up to
homotopy map K > L. In many cases the required naturality is too hard
to verify (or false) and so a second form of the theorem is used. Here
we simply suppose that the complex (Kn)G* > Kn > 0 is acyclic (an
easy consequence of the splitting above), that LG > L(1)G > 0 is
acyclic and that K(1) is isomorphic to L(1) and conclude that H(K) is
isomorphic to H(L). (Kn)G* stands for the standard powersofG
resolution coming from eps. This version is easy to apply, but suffers
from three defects. First, it works only in the case of isomorphism,
not arbitrary maps. Second, it does not in itself give naturality,
although that could probably be remedied by using a category of
relations. Third, and probably most important, it gives no uniqueness.
This means, for example, that although you can use it (in conjunction
with an argument involving simplicial subdivision) to show that singular
and simplicial homology of triangulated spaces are isomorphic, you
cannot show this way that the isomorphism is induced by the inclusion of
the simplicial chains into the singular ones.
I have recently discovered a version of acyclic models that repairs all
three defects. Moreover, it gives a single proof of both forms as well
as third form involving what I will call weak homotopy equivalence.
(This is not a Quillen model category in general, although there would
appear to be considerable overlap.) Let C be the category of
chain complexes of functors from some category X to an abelian category
A. Say that an arrow K > L in C is a weak homotopy equivalence if for
each object x of X, Kx > Lx is a homotopy equivalence (has a homotopy
inverse and homotopies, etc., but not assumed natural). Let Sigma stand
for one of the classes:
(a) homotopy equivalences
(b) weak homotopy equivalences
(c) homology isomorphisms
and let D denote the category of fractions gotten from C by inverting
Sigma. Let (G,eps) be a pair consisting of an endofunctor on X and a
natural transformation G > Id. Say that the augmented object K >
K(1) > 0 of C is Sigmatrivial if the 0 endomorphism is in Sigma.
Say that the object K of C is G presentable (w.r. to Sigma) if for each
n >= 0, the chain complex (Kn)G* > Kn > 0 is Sigmatrivial and K is
G acyclic (w.r. to Sigma) if KG > K(1)G > 0 is Sigmatrivial. Then
Theorem: If K is G presentable and L is G acyclic, both w.r. to Sigma,
then any natural transformation K(1) > L(1) can be extended in D to
an arrow, unique in D, K > L.
In case (a), this is the theorem of BarrBeck, 1966 and in case (c),
this repairs the three defects cited above, while case (b) appears to be
genuinely new.
The proof is embarrassingly easy. Consider the diagram
(alpha K)G* K(eps*)
K(1)G* < KG* > K




v (alpha L)G* L(eps*)
L(1)G* < LG* > L
alpha K and alpha L are the augmentation arrows. The Gpresentability
implies that K(eps*) is in Sigma and the Gacyclicity that (alpha L)G*
is. When these are inverted, we get the desired map K > L as the
composite. A paper on the subject will be posted in the usual ftp
location within a week or two.
From catdist@mta.ca Ukn Jan 16 12:53:10 1994
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Date: Sun, 16 Jan 1994 12:44:32 +0400 (GMT+4:00)
From: categories
Subject: Re: cantorbernstein
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Date: Sat, 15 Jan 94 11:36:22 EST
From: Michael Barr
One additional example and you don't even need retracts. In the category
of finitely generated modules over a commutative ring, all epis are
isos. As a result, if you have epis in both directions, they are
isos. So the dual category category is SB. This is fairly
easy if the ring has ACC, but there is a trick that works for any
ring to reduce it to that case. Since f.d. vector spaces are selfdual,
this example encompasses that one.
Michael
From catdist@mta.ca Ukn Jan 17 09:36:47 1994
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From: categories
Subject: Re: simple characterization of weak cartesian closedness
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Date: Mon, 17 Jan 94 9:44:41 MET
From: Simone Martini
Thomas Streicher asks
>> whether the following trivial observation is generally known:
>>
>>a category C with finite products is WEAKLY CARTESIAN CLOSED iff
>>for all objects A , B in C the functor C( _ x A , B) is a
>>retract of a representable functor.
I cannot say about "generally known", but..
this property it is quoted as one of the elementary characterizations of wCCC
in a paper of mine (Categorical Models for nonextensional lambdacalculi,
Mathematical Structures in Computer Science (1992), vol 2, pag 327357).
The paper, which has a definite didactic pace, discusses also the
case where there is only an epy natural transformation from C(_,A=>B) to
C( _ x A , B), which gives models of typed, non extensional, Combinatory
Logic; and it gives conditions on the existence of models of
typefree lambdacalculus as reflexive objects in wCCCs.
Simone Martini
Universit\`a di Pisa,
Dipartimento di Informatica.
From catdist@mta.ca Ukn Jan 17 09:40:11 1994
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Date: Mon, 17 Jan 1994 09:29:33 +0400 (GMT+4:00)
From: categories
Subject: Re: simple characterization of weak cartesian closedness
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Date: Mon, 17 Jan 1994 11:37:33 +0100 (MET)
From: Raymond Hoofman
Quoting Thomas Streicher,
> I wonder whether the following trivial observation is generally known
> :
>
> a category C with finite products is WEAKLY CARTESIAN CLOSED iff
> for all objects A , B in C the functor C( _ x A , B) is a
> retract of a representable functor.
Yes, this is the "degenerate" case of a semiadjunction between a functor G
and a semifunctor F: the semiisomorphism
D(F(), ...) \cong_{s} C(, G(...))
becomes a retraction (see [1], also [2]).
> I don't think that the remark above is a deep insight !!
> BUT usually people refer to the quite heavy machinery of Hayashi's
> semifunctors when they speak about the categorical semantics of typed
> lambda calculus without etarule.
However, if the products of your typed lambda calculus also do not satisfy
the etarule, the semiisomorphism above does not degenerate, and it is
less obvious how to give a simple characterization without semifunctors (apart
from saying that the Karoubi envelope of the category is Cartesian closed).
[1] The theory of Semifunctors, R. Hoofman, MSCS 3
[2] Collapsing Graph models by preorders, R. Hoofman & H. Schellinx, LNCS 530
With kind regards,
Raymond Hoofman.
From catdist@mta.ca Ukn Jan 17 16:49:30 1994
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From: categories
Subject: Fuzzy references
To: categories
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Date: Mon, 17 Jan 1994 12:57:49 0400
From: Mike Wendt
==========================================================================
Hi Al:
I'm not sure if this is what you want but here are a couple of references
I've noticed recently (my search for categoricalmeasuretheorytype stuff):
Bandemer, H., Nather, W, "FUZZA DATA ANALYSIS," Theory and Decision Library,
Kluwer Academic Press, Series B, Vol. 20 (Norwell, Mass., 1992).
Rodabaugh, S., Klement, E., Hoehle, U. (eds.), "APPLICATIONS OF CATEGORY
THEORY TO FUZZY SUBSETS," Kluwer Academic Press, Series B, Vol. 14
(Norwell, Mass., 1992).
I'm sorry, I can't give you a review of these books yet. I have only peeked
in the first one. It seems interesting enough and is at an introductory
level.
Regards,
Mike Wendt
==========================================================================
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From: categories
Subject: Re: Fuzzy +
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Date: Mon, 17 Jan 94 15:36:35 +0100
From: Pierre Ageron
In my thesis "Structure des logiques et logique des structures", I tried (very
shortly) to understand what the algebraic and categorical counterparts of
fuzzy logic are.
There are different proposals in the literature, the reason is that there
are plenty of interesting operations on the interval [0,1] and that it is very
difficult to tell which ones are relevant for fuzzy logic. The most general
axiomatization was given by Rene Guitart in his 1979 thesis (or a 1982
paper in the Cahiers). He considered complete ordered abelian monoids (in
their 1990 book, Barr and Wells restricted to complete Heyting algebras).
I observed that every complete ordered abelian monoid has a canonical
Lafont algebra structure: this means that (this) fuzzy logic is the extension
of intuitionistic linear logic with infinitary versions of the additive
connectives "plus" and "with".
Guitart defined the notion of "algebraic universe": essentially a category
equipped with a monad P looking like the monad of subsets on Ens (I mean
Set !). This notion subsumes the notion of elementary topos and allows
to give higher order semantics for logics other than intuitionistic logic.
In the case of fuzzy logic, the point is that every complete ordered abelian
monoid defines such a structure on Ens. The Kleisli category of P is the
category of fuzzy relations. All this is explained in my thesis using
the notations of linear logic.
All that framework gives Tarskian semantics for (propositional or
higher order) fuzzy logic. It is not clear whether there are Heytingian
semantics for fuzzy logic, i.e. a proof theory. The difficulty is that
every small complete category is a poset (but this result by Freyd uses
AC, so hope remains...)
Pierre AGERON
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From: categories
Subject: Re: cantorbernstein
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Date: Mon, 17 Jan 94 10:13:02 EST
From: Peter Freyd
Mike Barr writes:
One additional example and you don't even need retracts. In the category
of finitely generated modules over a commutative ring, all epis are
isos. As a result, if you have epis in both directions, they are
isos. So the dual category category is SB. This is fairly
easy if the ring has ACC, but there is a trick that works for any
ring to reduce it to that case.
Wonderful thought: all epis are isos. Anyway, I see a proof that any
epi endo on a finitely presented module over a commutative ring is
iso, but finitely generated?
There's a metaprinciple that says that a result like this should
generalize from commutative to PI rings (that is, rings that satisfy
some nontrivial Polynomial Identity). Can anyone confirm? A corollary
would be that in any additive category if two objects each appear as
retracts of the other, and if the ring of endomorphisms of one of them
is a PI ring then the retractions are isos.
best thoughts,
peter
From catdist@mta.ca Ukn Jan 17 22:08:34 1994
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From: categories
Subject: mathematics made hard
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From: Peter Freyd
This is about the opposite of category theory. I'm going to give
a soft proof of something and ask how to get the hard proof. All
of this because of Mike Barr's note about modules of commutative rings.
Let M be a finitely presented module over a commutative ring R
and f:M > M an epimorphic endomorphism. We will show that
f is necessarily an isomorphism. First specialize to the case that
R is Noetherian. The kernels of the powers of f form an ascending
chain of submodules of M, hence must stabalize. That is, there
k+1 k k+1
is a natural number k such that Ker(f )= Ker(f ). Since f is
epi, it is a cokernel for its kernel and there must exist g:M > M
k+1 k
such that f g = f . (I'm composing maps in the diagramatic order.)
Using for the second time that f is epi we may cancel to obtain
fg = 1. Since fgf = f1 we cancel once more (using that f is epi
for the third time) to obtain gf = 1.
Now, let r and n be natural numbers and
r n
R > R > M > O
an exact sequence. There must be an rxn matrix K, an rxr matrix
A' an nxn matrix A, another nxn matrix B, and an nxr matrix
C such that
KA = A'K
BA + CK = I. (K describes the map
r n n
from R to R that defines M, A describes the endomorphism on R
r
that "lifts" f, A' describes the endomorphsim on R . Since f is
n+r n
epi the map R > R obtained by stacking A and K is also
n n+r
epi, hence it has a leftinverse (B,C):R > R .)
Specialize to the case that R is the the "generic ring",
that is the ring generated by the 2nn+2nr+rr entries of K,A,A',B,C
with nr+nn equations. We may infer that there is an
rxr matrix X and an nxr matrix Y such that
KB = XK
AB + YK = I.
The entries of X and Y are necessarily given by polynomials in
the generating "variables" and the last two matrix equations must
result in rr+nr equations that are direct consequences of the
nr+nn defining equations. Hence the original theorem works for any
finitely presented module over any commutative ring.
Now for the hard part: what are these polynomials? In the case n = 1
its easy (and reveals quickly the need for commutativity).
Try it for n=2, r=1. Given:
ac+be = ga
ad+bf = gb
hc+ie+la = 1
hd+if+lb = 0
jc+ke+ma = 0
jd+kf+mb = 1
find, for a start, a polynomial on these variables, x, such that
ah+bj = xa
ai+bk = xb.
From catdist@mta.ca Ukn Jan 17 22:18:57 1994
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From: categories
Subject: Re: cantorbernstein
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Date: Mon, 17 Jan 94 20:15:09 EST
From: Michael Barr
> Mike Barr writes:
>
> One additional example and you don't even need retracts. In the category
> of finitely generated modules over a commutative ring, all epis are
> isos. As a result, if you have epis in both directions, they are
> isos. So the dual category is SB. This is fairly
> easy if the ring has ACC, but there is a trick that works for any
> ring to reduce it to that case.
>
> Wonderful thought: all epis are isos. Anyway, I see a proof that any
> epi endo on a finitely presented module over a commutative ring is
> iso, but finitely generated?
>
> There's a metaprinciple that says that a result like this should
> generalize from commutative to PI rings (that is, rings that satisfy
> some nontrivial Polynomial Identity). Can anyone confirm? A corollary
> would be that in any additive category if two objects each appear as
> retracts of the other, and if the ring of endomorphisms of one of them
> is a PI ring then the retractions are isos.
>
> best thoughts,
> peter
>
>
I will try to recall the argument (online).
Given an epiendomorphism f, look at the ascending chain ker(f),
ker(f^2), ker(f^3),.... In the noetherian case, this stabilizes so
that ker(f^n) = ker(f^{n+1}). Assume thatn is as small as possible,
so that ker(f^{n1}) < ker(f^n). Choose an element x in the ker of
f^n, not in the lesser one. x = f(y) for some y, since f is onto.
0 = f^n(x) = f^{n+1}(y), so that 0 = f^{n}(y) = f^{n1}(x), a
contradiction. That takes care of the noetherian case and doesn't
even use commutativity, it would seem. For the general case,
suppose R is the ring, M the module, f: M > M the endomorphism and
x an element with f(x) = 0. Now pick a set of generators for M, say
y_1,...,y_n. What you have to do is to find a suitable finite
subset of R, with just the right elements in it to express all the
f(y_i), x and at least one preimage of each y_i as linear
combinations of the y_i using coefficients from that subset. Now
let S be the subring of R generated by that finite set of elements
and N be the least Ssubmodule of M containing all the y_i. If I
have left anything required out of S, add that too. Anyway, S is
noetherian (this does use commutativity, I believe) and f induces a
counterexample on N.
I believe this argument is due to one of the Rutgers people like
Faith or Osofsky, but I am far from certain of that. It will be
true for PI rings if affine PI rings have acc on left ideals. For
commutative rings it is essentially the Hilbert basis theorem.
Michael
From catdist@mta.ca Ukn Jan 19 22:39:16 1994
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From: categories
Subject: RE Fuzzy +
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Date: Wed, 19 Jan 1994 11:45:07 0600
From: Lawerce Neff Stout
I've done quite a lot of work on categories of fuzzy sets. The main paper
is in the volume edited by H\"ohle and Rodabaugh referred to earlier.
Barr is correst that fuzzy sets form a quasitopos, but the logic of that
quasitopos is that of the underlying set category, hence not interesting
as a place to do fuzzy mathematics. The fuzzy connectives come from a
second monoidal closed structure obtainable from, for example, the tnorms
usually referred to in the fuzzy literature. This gives a very satisfactory
logic if one uses what I called unballanced subobjects (the map involved is
both monic and epic). There is a weak representor for these subobjects
(representation is not unique though there is an ordering on maps which
allows a canonical choice of representative to be made) allowing an internal
representation of a large fragment of higher order fuzzy logic.
I have a more recent paper (to appear in the proceedings of the 1992 Linz
seminar, being published by Kluwer sometime later this year) in which I look at categories of fuzzy sets with values in a Quantale or Projectale. That paper is
available from me by email (I don't have ftp facilities available). It
includes a characterization of categories of fuzzy sets in terms of the
representability of the logic and the property of being topological over
Sets.
Larry Stout
From catdist@mta.ca Ukn Jan 21 09:51:34 1994
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From: categories
Subject: categorical treatment of F_omega?
To: categories
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Date: Thu, 20 Jan 1994 17:24:00 0500
From: David Espinosa
1. Could someone send me a good reference for a categorical treatment
of the Girard / Reynolds F_2 polymorphic type system? That is,
polymorphic functions as (some form of) natural transformations?
2. More importantly, has there been a categorical treatment of
Girard's F_omega type system?
David
From catdist@mta.ca Ukn Jan 22 13:58:28 1994
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Date: Sat, 22 Jan 1994 13:49:21 +0400 (GMT+4:00)
From: categories
Subject: Terminology question
To: categories
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Date: Sat, 22 Jan 1994 2:44:17 0500 (EST)
From: D_FELDMAN@UNHH.UNH.EDU
Is there a standard terminology for the following sort of gadget or something
very similar?
Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and
a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite
sets and bijections, satisfying\\
(i) If $F(s_1)=t_1$ and $\tau:t_1\rightarrow t_2$, then there exists a
unique object $s_2\in {\bf S}$ and a unique ${\bf S}$morphism
$\sigma:s_1\rightarrow s_2$ such that $F(\sigma)$=$\tau$.\\
(ii) For $t \in {\bf Bij}$, $F^{1}(t)$ is an (effectively
computable) finite set.
David Feldman
From catdist@mta.ca Ukn Jan 24 13:43:31 1994
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From: categories
Subject: Re: Terminology question
To: categories
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Date: Mon, 24 Jan 1994 16:13:07 +0000 (GMT)
From: Edmund Robinson
>
> Date: Sat, 22 Jan 1994 2:44:17 0500 (EST)
> From: D_FELDMAN@UNHH.UNH.EDU
>
> Is there a standard terminology for the following sort of gadget or something
> very similar?
>
> Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and
> a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite
> sets and bijections, satisfying\\
> (i) If $F(s_1)=t_1$ and $\tau:t_1\rightarrow t_2$, then there exists a
> unique object $s_2\in {\bf S}$ and a unique ${\bf S}$morphism
> $\sigma:s_1\rightarrow s_2$ such that $F(\sigma)$=$\tau$.\\
> (ii) For $t \in {\bf Bij}$, $F^{1}(t)$ is an (effectively
> computable) finite set.
>
> David Feldman
>
>
I think this would traditionally be described as a "finite discrete
opfibration (over Bij)". The functors corresponding to condition (i)
are discrete opfibrations, and the finite comes from condition (ii).
Neither of these uses any special property of Bij (such as the fact
that it is a groupoid). It might be more modern to use "cofibration"
instead of "opfibration". See Barr & Wells "Toposes, Triples and
Theories" p231 ex [OPF] for more conventional definitions, and perhaps
Benabou "Fibred categories and the foundations of naive category
theory" (J. Symbolic Logic (50) No. 1, 1985, 1037) for more of an
indication of why these sorts of structures are so common.
Another way of looking at the structure would be to turn it around and
say that you have a functor G: Bij > FiniteSet given on objects by
G(t) = F^{1}(t).
best wishes,
Edmund Robinson
From catdist@mta.ca Ukn Jan 25 07:26:29 1994
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Date: Tue, 25 Jan 1994 07:21:14 +0400 (GMT+4:00)
From: categories
Subject: Re: Terminology question
To: categories
Cc: cdl2  France Dacar ,
Robert Dawson ,
"Oege de.Moor" ,
"Valeria de.Paiva" ,
"Ruy de.Queiroz" ,
"FerJan De.Vries" ,
KyungGoo Doh ,
James Dolan ,
Xiaomin Dong ,
Winfried Drecmann ,
Dominic Duggan ,
Gerald Dunn , Hans Dybkjaer ,
Abbas Edalat ,
David Espinosa ,
Michel Eytan ,
Joe Fasel , David Feldman ,
Zbigniew Fiedorowicz ,
Juarez Muylaert Filho ,
Stacy Finkelstein ,
Kathleen Fisher ,
Maria Frade , Peter Freyd ,
Tom Fukushima ,
Jonathan Funk ,
Fabio Gadducci ,
Vijay Gehlot ,
Wolfgang Gehrke ,
Silvio Ghilardi ,
Paul Glenn , Joseph Goguen ,
Marek Golasinski ,
Al Goodloe ,
Bob Gordon ,
Francoise Grandjean ,
John Gray , Luzius Grunenfelder ,
Stefano Guerrini ,
Alessio Guglielmi ,
James Harland , Robert Harper ,
Magne Haveraaen ,
"Michael J. Healy" ,
Michel Hebert ,
Murray Heggie ,
Luis Javier Hernandez , Walt Hill ,
SATO Hiroyuki ,
Bernard Hodgson
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Date: Tue, 25 Jan 1994 10:18:25 +0000
From: Steven Vickers
Do others suffer the same heartsink as I do when confronted with a posting
like this?
>Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and
>a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite
>sets and bijections, satisfying\\
> (i) If $F(s_1)=t_1$ and $\tau:t_1\rightarrow t_2$, then there exists a
>unique object $s_2\in {\bf S}$ and a unique ${\bf S}$morphism
>$\sigma:s_1\rightarrow s_2$ such that $F(\sigma)$=$\tau$.\\
> (ii) For $t \in {\bf Bij}$, $F^{1}(t)$ is an (effectively
>computable) finite set.
For human readers (and after all, is this message _ever_ going to be
presented to a Latex interpreter?) most of the $'s and \'s here are not
only completely unnecessary, but, worse, a positive barrier to
understanding.
I would expect  but this is something that can be put to the test  that
even people completely familiar with Latex would find it easier to read the
following version. It certainly involves less typing.
>Define a ?????? to be a pair (S,F) consisting of a category S and
>a functor F from S to Bij, the category of finite
>sets and bijections, satisfying 
> (i) If F(s_1)=t_1 and tau:t_1 > t_2, then there exists a
>unique object s_2 in S and a unique Smorphism
>sigma: s_1 > s_2 such that F(sigma)=tau.
> (ii) For t in Bij, F^{1}(t) is an (effectively
>computable) finite set.
(I have ignored the puzzle of whether {\bf ...} is mathematically
meaningful  in the original $S$ turns into ${\bf S}$. If it _is_
mathematically meaningful, then in Latex it should be macroized.)
Steve Vickers.
p.s. Having rephrased the question, I still don't know the answer  sorry.
From catdist@mta.ca Ukn Jan 25 12:17:27 1994
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Date: Tue, 25 Jan 1994 11:47:26 +0400 (GMT+4:00)
From: categories
Subject: Re: cantorbernstein
To: categories
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Date: Wed, 19 Jan 1994 09:37:14 0500
From: Stephen Chase
With regard to the remarks of Barr and Freyd on surjective endomorphisms
of finitely generated modules: I haven't digested those remarks, but here
is a slick proof of the result, communicated to me years ago by Bill
Waterhouse (instead of reduction to the Noetherian case, it uses
localization):
It is enough to prove bijectivity at all localizations, so we can assume
that the commutative ring A is local with maximal ideal m. Given a
surjective endomorphism f of a finitely generated Amodule M, let F be a
finitely generated free Amodule mapping onto M so that the mapping
induces an isomorphism F/mF > M/mM. Let K = Ker(F > M). f then
lifts to an endomorphism g of F, which is an isomorphism because it is
so mod m. Then g(K) is contained in K, and to prove f is bijective we
need only show g'(K) is likewise contained in K (with g' the inverse of g).
But g satisfies its characteristic polynomial, which has invertible constant
term det(g) (up to sign); thus g' is a polynomial in g and so maps K into
itself.
I haven't seen this proof in the literature. However, the following
related reference might be of interest: M. Orzech, L. Ribes, "Residual
finiteness and the Hopf property in rings", J. Algebra 15 (1970), 8188.
Sincerely,
Steve Chase
From catdist@mta.ca Ukn Jan 25 12:22:48 1994
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Date: Tue, 25 Jan 1994 11:53:49 +0400 (GMT+4:00)
From: categories
Subject: terminology
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[Note from moderator:
1. The two posts following are being forwarded, but I don't feel that this
list is really the place for a long discussion of suitable notation for
email, so I hope that any discussion will be short]
Date: Tue, 25 Jan 94 08:37:53 EST
From: Peter Freyd
I certainly agree with Steve's point. But I would go further:
Define a ?????? to be a pair (*S*, F) consisting of a category *S* and
a functor F from *S* to *Bij* , the category of finite
sets and bijections, satisfying 
(i) If F(S ) = T and f:T > T' , then there exists a
unique object S' in *S* and a unique morphism
g: S > S' such that F(g) = f.
(ii) For T in *Bij*, F (T) is an (effectively
computable) finite set.
But: I must confess that I also experience a little "heartsink" when I
see a list of addresses as long as that above.
best thoughts,
peter
++++++++++++++++++++++++++++++++++++++++++++
Date: Tue, 25 Jan 94 9:51:20 EST
From: Al Vilcius
Referring to the "rephrased" question of Steve Vickers:
>
> Do others suffer the same heartsink as I do when confronted with a posting
> like this?
>
> >Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and
> >a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite ......
Yes, I certainly do, and much prefer the "humanized" alternative:
>
> >Define a ?????? to be a pair (S,F) consisting of a category S and
> >a functor F from S to Bij, the category of finite ......
>

/\ / Al Vilcius, Toronto
/ \ /
/>\ /
/ \/
From catdist@mta.ca Ukn Jan 27 12:30:09 1994
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From: categories
Subject: RE: terminology
To: categories
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Date: Wed, 26 Jan 1994 2:35:50 0500 (EST)
From: D_FELDMAN@UNHH.UNH.EDU
Thank you to all those who responded, including those who pointed
out my efaux pas. Incidently, the complaint about S versus {\bf S}
alerted me to a typo in a paper under preparation (these should have
been the same) and so I am especially grateful for that.
David Feldman
From catdist@mta.ca Ukn Jan 31 16:29:46 1994
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From: categories
Subject: New address of FerJan de Vries
To: categories
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Date: Mon, 31 Jan 1994 20:17:52 +0100
From: F.J.de.Vries@cwi.nl
CWI, January 31, 1994
Dear Colleague.
The coming year I will live and work in Japan.
My addresses will be the following:
Office: from March 1st, 94, onwards
NNT,
Communication Science Laboratories
Hikaridai, Seikacho,
Sorakugun, Kyoto 61902
Phone +81774951841,
Facsimile +81774951851
Email ferjan@progn.kecl.ntt.jp
Home: from Feb 1st, 94, onwards
SeresuGakuenmae 305
GakuenNaka 11542190
Narashi, Nara 631
Phone: yet unkown...
Sayonara, FerJan de Vries.