Jean Bénabou | 30 Jul 03:05 2014
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Re: A brief survey of cartesian functors

Dear George,

Thank you for your mail. I see that all my mathematical arguments have not convinced you, and that trying to
add more would be useless.
I respect your opinion although I totally disagree with it.

Best regards,
Jean

Le 29 juil. 2014 à 21:58, George Janelidze a écrit :

> Dear Jean,
> 
> Thank you for your kind words at the beginning of your message, and I apologize if what I said about
"factorization" and "cartesian" was unclear.
> 
> I did not mean to say that there is any connection between factorization systems and (pre foliations +
cartesian FUNCTORS). What I was trying to say, was only that the following two constructions are
essentially the same (up to an isomorphism):
> 
> (a) For a fibration C-->X every morphism f in C factors as f = me, where m is a cartesian ARROW and e is a
vertical arrow (with respect to the given fibration).
> 
> (b) For a semi-left-exact reflection C-->X (in the sense of Cassidy--Hebert--Kelly) every morphism f in
C factors as f = me, where m is in M, e is in E, E is the class of all morphisms inverted by C-->X, and M is its
orthogonal class (M can also be defined as the class of trivial covering morphisms in the sense of Galois theory).
> 
> I know this might sound trivial to you, but I think it is a fundamental connection, which should be widely
known. And I believe that instead of
> 
(Continue reading)

Tom Leinster | 29 Jul 22:50 2014
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Re: New book: Basic Category Theory

Dear Harley,

Thanks!

Actually, some of us were just discussing the possibility of writing a
second category theory book as a collaborative venture:

http://golem.ph.utexas.edu/category/2014/07/basic_category_theory.html#c046968

(But personally, I've probably had enough book-writing for a while.)

Best wishes,
Tom

On Tue, 29 Jul 2014, Harley D. Eades III wrote:

> Hi, Tom.
>
> Congrats on the successful completion of your book!
>
> I really like your idea on a second book.  Such a book would be
> very helpful for young researchers and students in my opinion.
>
> Has anyone thought about doing it as a group effort in the same
> spirit as the homotopy type theory book was?
>
> A number of authors would make the writing burden far less per author,
> and potentially speed up the writing process.
>
> Anyway, just an idea.
(Continue reading)

George Janelidze | 29 Jul 21:58 2014
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Re: A brief survey of cartesian functors

Dear Jean,

Thank you for your kind words at the beginning of your message, and I
apologize if what I said about "factorization" and "cartesian" was unclear.

I did not mean to say that there is any connection between factorization
systems and (pre foliations + cartesian FUNCTORS). What I was trying to say,
was only that the following two constructions are essentially the same (up
to an isomorphism):

(a) For a fibration C-->X every morphism f in C factors as f = me, where m
is a cartesian ARROW and e is a vertical arrow (with respect to the given
fibration).

(b) For a semi-left-exact reflection C-->X (in the sense of
Cassidy--Hebert--Kelly) every morphism f in C factors as f = me, where m is
in M, e is in E, E is the class of all morphisms inverted by C-->X, and M is
its orthogonal class (M can also be defined as the class of trivial covering
morphisms in the sense of Galois theory).

I know this might sound trivial to you, but I think it is a fundamental
connection, which should be widely known. And I believe that instead of

"indexed categories versus fibrations"

one should sometimes also consider

"indexed categories versus fibrations versus semi-left-exact reflections"
(this is why I mentioned a "third approach").

(Continue reading)

Jean Bénabou | 29 Jul 11:16 2014
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Re: A brief survey of cartesian functors

Dear George,

I appreciate very much your pioneer work on Galois theories and the developments you and others have given
to that work.
I also believe in the role of analogies in mathematics, and I think category theory is the ideal place where
one can give the DEEP analogies a mathematical content.
However, in this case, the analogy seems to me totally superficial, namely: two classes A and B of maps in a
category X, and the possibility to factor every map f of X as ab, with a in A and b in B. 

This won't go very far since you need some axioms on the pair (A,B) to start proving anything except
trivialities. And, I tried to explain in my previous mail, the properties of pairs (E,M) and (V,K) are so
radically different that a common denominator would be reduced to almost nothing.

Even more important to me, cartesian functors are a very good notion of morphism between pairs (V,K) and
(v',K')  which you can prove non trivial results, the theorem in my mail is only an example of such results.
As far as I know there is no notion of morphism between pairs (E,M) and (E',M').

Let me point out some features of cartesian functors F: X --X' , viewed abstractly as morphisms (V,K) -->
(V',K')  where V = V(P),  K = K(P),  V' = V(P') and K' = K(P').
1) F preserves vertical end cartesian maps. This is harmless, but F also REFLECTS vertical maps.
2) We assume that every map of X can be factored as kv, but we make no such assumption on X'
3) The very nature of the results: For any important properties, F satisfies globally the property iff it
satisfies it fiberwise.

If any reasonable notion of morphism of pairs (E,M) was defined someday would reflection of maps in M be
considered? Would one accept that (E',M') should not be a factorization system even in a very weak sense?
And if non trivial results could be obtained about such notion would some kind of fibers play a role?

Sorry George, much as I like unifying notions and theories, I cannot see any real, non trivial, relation
between factorization systems and (pre folations + cartesian functors)
(Continue reading)

George Janelidze | 29 Jul 09:02 2014
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Re: A brief survey of cartesian functors

Dear Jean,

Talking about the comparison, I had in mind mainly the following: the
vertical-cartesian factorization for a fibration is closely related to the
reflective factorization system for a semi-left-exact reflection (one might
vaguely say "they are the same up to an isomorphism under the assumptions
used in both of them").

Concerning the older discussion on fibrations versus indexed categories:
Please believe me that I fully agree with every instance of "fibrations are
better" you mention. Nevertheless I also agree with "indexed categories are
better", in a different sense. The reason I am saying this now is that I
would like to mention semi-left-exact reflections of Cassidy--Hebert--Kelly
and their generalizations as a THIRD APPROACH (I used them independently
calling them "admissible" in Galois theory, first exactly in 1984).

Best regards,
George

--------------------------------------------------
From: "Jean B?nabou" <jean.benabou <at> wanadoo.fr>
Sent: Monday, July 28, 2014 1:58 PM
To: "George Janelidze" <janelg <at> telkomsa.net>
Cc: "Ross Street" <street <at> ics.mq.edu.au>; "Steve Vickers"
<s.j.vickers <at> cs.bham.ac.uk>; "Lack Steve" <steve.lack <at> mq.edu.au>; "Peter
Johnstone" <P.T.Johnstone <at> dpmms.cam.ac.uk>; "Eduardo Dubuc"
<edubuc <at> dm.uba.ar>; "Thomas Streicher"
<streicher <at> mathematik.tu-darmstadt.de>; "Robert Par?"
<pare <at> mathstat.dal.ca>; "Marta Bunge" <martabunge <at> hotmail.com>; "William
Lawvere" <wlawvere <at> hotmail.com>; "Michael Wright" <mpbw1879 <at> yahoo.co.uk>;
(Continue reading)

Keith Harbaugh | 29 Jul 00:05 2014
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Les Distributeurs as a TAC Reprint

Many papers on category theory cite
Jean Bénabou, Les distributeurs, Université Catholique de Louvain,
Institut de Mathématique Pure et Appliquée, rapport 33, 1973.
For some reason (hint: anybody know why?)
this seems never to have been published.
Would it not be a good idea to publish it as a TAC Reprint,
in view of both
its current scientific/mathematical value and
its value as a seminal source for many subsequent developments?

As an example of what is being said about it on the web,
the following is quoted from the nlab page on profunctors as of this date:

The original published source for profunctors is
Jean Bénabou, Les distributeurs, Université Catholique de Louvain,
Institut de Mathématique Pure et Appliquée, rapport 33, 1973
based on several series of lectures starting in 1969, but these notes
are hard to come by. They are available from the author by request.
Much (if not most) of the existing work on profunctors has been
developed by Bénabou.

I don't know the politics of getting it reprinted,
but simply on scientific and historical grounds,
it seems that would be a very good idea,
and one I am by this email requesting.
Further, perish the thought, we are all mortal,
and the issue arises how future generations might have access to
a work that is cited so frequently.

Best, Keith
(Continue reading)

Michael Barr | 28 Jul 22:35 2014
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Site for my papers

For a number of years, I made most of my papers available at an ftp site 
at McGill which is now defunct.  It was also not always up-to-date.  I 
have a new one that is, for the the being, up-to-date.  If you enter 
http://www.math.mcgill.ca/barr/papers/ in your browser, it will take you 
to the index, currently by production date.  Eventually, I will add a 
subject index as well.  Right now it includes most of my lifetime 
production (except most of what was published in Lecture Notes) and a few 
other things such as the Grothendieck translation and some notes on rings 
of continuous functions by Fine, Gilman, and Lambek from about 1960 that 
was published by McGill with no copyright notice (anyway, I got Lambek's 
and Gillman's permissions (but Fine was dead and I could not trace his 
widow or his daughter).  My only connection was that I supervised the 
texxing of the notes.

Michael

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Jean Bénabou | 28 Jul 19:36 2014
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Re: A brief survey of cartesian functors

Dear André

Your guess is quite correct. More generally every full and faithful functor is a foliation. 
Let me call a functor P: X --> S locally full and faithful (lff)  iff for each object  x of X the obvious functor 
X/x --> S/P(x)  is full and faithful. Such functors are also characterized by :  Every map of X is
hypercartesian.  Thus they are foliatiions. Now every full and faithful functor is lff. Hence is a foliation.
I mentioned in my mail that there are many foliations which are not fibrations, this is a typical example. It
shows how much more general than (pre) fibrations  (pre) foliations can be.
To give an easy but important application, let me note that, if X is a groupoid avery functor  P: X --> S, where S
is arbitrary, is a foliation, because all the maps of X are isos, hence hypercartesian.

This gives me the opportunity to explain condition (ii) for cartesian functors in a special case. Suppose
P: X --> S,  P' :  X' --> S and F; X --> X'  verify  P = P'F,  where X, X'  and S are groups. Then P and P' are foliations and F
preserves cartesian maps. However F need not be cartesian. More precisely F satisfies (ii) iff P and P'
have same image in S. In that case the theorem says:  F is a mono (resp an epi)  iff its  restriction  Ker(P) -->
Ker(P') is a mono (resp an epi). This would be obviously false without (ii) 
To complete the picture let us see that  (i) => (ii)  when P is a fibration. In that case P is surjective, i.e.
Im(P) = S  contains Im(P') . But  P = P'F => Im(P) is contained in Im(P'), hence the equality required.

Thank you for having given me the occasion to explicit some examples, an in particular to show that (ii) is meaningful.

Bien amicalement,
Jean

Le 28 juil. 2014 à 17:53, Joyal, André a écrit :

> Dear Jean,
> 
> I apologise for my ignorance of your work.
> 
(Continue reading)

Paoli, Simona (Dr. | 28 Jul 16:33 2014
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Lectureship at the University of Leicester

Dear Colleagues,

   a lectureship in Pure Mathematics is advertised at the University of Leicester (UK), see details below.

Best regards,

Simona.

Lecturership in Pure Mathematics

Department of Mathematics - University of Leicester (UK)

The University of Leicester wishes to appoint a full-time, open-ended Lecturer in Pure Mathematics. 

We are looking especially for applications in areas of pure mathematics complementing the exiting
research interests in the department.

Informal enquiries are welcome and should be made to the Head of Department  Professor Ruslan Davidchack on
rld8 <at> le.ac.uk or 0116 252 3819.

The closing date for this post is midnight on 3 September 2014.

More information about this position and how to apply can be found here:
http://www.jobs.ac.uk/job/AJG336/lecturer-in-pure-mathematics/

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Jean Bénabou | 28 Jul 16:13 2014
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The paper you sent me

Dear George,

As I expected, the paper deals with factorization systems (E, M).
It has NOTHING ,  but NOTHING to do with pre foliation, foliations, pre fibrations and fibrations, let alone
with cartesian functors. here are a few reasons, I could give many many more.
Let me call for short a cartesian system a pair (V, K) where V is the set of vertical maps and K the set of
cartesian maps (in the sense of Grothendieck, which I recalled in my mail) of a pre foliation P: X --> S .
You can, if you prefer, assume that P is foliation, a pre fibration, or even a fibration. The following
remarks work in all cases.
.
1.  In (E, M) all the isos are both in E and M. In (V, K) the isos need not be in V. In most important cases they will
NOT be in V.
2.  In (E, M) each of the classes E and M determines the other.  In (V,K)  V determines K, but K does not determine
V. The most extreme case is the following. Suppose X is a groupoid. For every P: X -> S,  with S arbitrary every
map of is cartesian (an even hyper cartesian) i.e K(P) =  X, hence P is a foliation.
Take P = id : X --> X  and  Q: X --> 1 the unique functor. They are both fibrations, K(P) = K(Q) = X , but V(P) consists
only of the identities of X whereas V(Q) is the whole of X
As a side remark factorization systems don't make any sense on a groupoid, there is only one, whereas 
foliations or fibratons with domain a groupoid make perfect sense and are even very important.
3. In (E, M) both E and M are stable by composition, whereas K need not be if P is a pre foliation or a pre fibration
4. For every functor P, V(P) satisfies 3 out of 2, i.e. for every commutative triangle in X, if 2 of the maps are
vertical so is the third. This is almost never the case for factorization systems.
5. Factorizations in (E, M) are functorial, they need not be in (V,K) for pre foliations , or even pre fibrations
How much more do you need?

Let me add that in the paper you sent me the authors assume that the category A, where (E, M) lives admits
finite limits and all intersections (even large ones) of strong subobjects. This assumption is totally
irrelevant in my work on fibrations or (pre) foliations

For cartesian functors the situation is even more hopeless. They are special morpphisms in Cat/S. As far as
(Continue reading)

Jean Bénabou | 28 Jul 13:58 2014
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Re: A brief survey of cartesian functors

Dear George, 

As i mentioned in my mail, it took me many years to develop foliated categories AND cartesian functors to
their full extent. The very first approach starts circa 1984 when I had proved important results on
cartesian functors between fibered categories and started wondering about possible generalizations.
Thank you for sending me the paper of Cassidy, Herbert and Kelly which I do not know. I shall look at it
carefully, but I doubt very much that it will have ANYTHING to do with foliated categories, let alone
cartesian functors which are the essential content of my mail.
I shall explain why after I have read the paper you are sending me.

Best regards to all,
Jean

Le 28 juil. 2014 à 12:52, George Janelidze a écrit :

> Dear Jean,
> 
> I remember you talking about foliations more than 20 years ago, but when exactly is this done? Long before?
> 
> No matter what was done first, I think it would nice to compare this carefully with the results of
> 
> [C. Cassidy, M. Hébert, and G. M. Kelly, Reflective subcategories, localizations, and factorization
systems, Journal of Australian Mathematical Society (Series A), 1985, 287-329].
> 
> The seemingly big difference is that the above-mentioned paper is about reflections, but in fact having
the right adjoint is a much weaker restriction than it seems (in this context).
> 
> Since we don't sent attachments to Categories mailing list, I shall send you that paper separately.
> 
> With best regards to all,
(Continue reading)


Gmane