Re: Bourbaki and Categories

Re: Bourbaki and Categories

[Steve Lack]

Dear Andrei,

Sketches are not mathematical objects in their own right, in the same sense
that groups or spaces are. They are presentations (for theories), and have
status similar to other sorts of presentations (for groups, rings, etc.)

Of course that is in no way meant to suggest that they are not important and
worthy of study.

Regards,

Steve Lack.


On 16/09/08 11:09 PM, "Andre.Rodin@ens.fr" <Andre.Rodin@ens.fr> wrote:

>
> Of course, you are right about a point, I missed it! I must confess I didn't
> think about this example in precise terms. My claim is that sketch theory
> doesn't fit the structuralist (Bourbaki-Hilbertian) pattern. It hardly
> precisely fits the ancient Euclidean pattern either but there is a suggestive
> analogy, which concerns the idea that certain basic objects like point, line
> and circle *generate* the rest.
> A further claim is this: a specific reason *why* sketch theory doesn't fit the
> structuralist pattern is that in sketch theory (like in CT in general)
> isomorphisms don't have the same distinguished status.
>
> andrei
>
>
>
>
>
>> I don't know what to say about the suggestion that a circle and a line
>> make a sketch of which Euclidean plane geometry is a model.  I would think
>> you would need a point too, since intersections are crucial.  Maybe
>> complex projective geometry since then two lines intersect in one point
>> (unless they coincide), a line and a circle in two (unless they are
>> tangent or equal) and every pair of circles in four (ditto).  Maybe the
>> exceptions could be handled in some sketch.  At any rate, it wold e
>> interesting to try to sketch this in detail.  At any rate, I never thought
>> about this before.
>
>> Michael
>
>
>

Re: Bourbaki and Categories

[jim stasheff]

I would just add or emphasize if implicit here one thought:

a view into the often very human process of interaction
viz. several of the anecdotes about `life with Bott' at
his recent memorial conference

jim



Meredith Gregory wrote:
> All,
>
> i have been utterly delighted by this conversation. What i can't help but
> think about, however, is that with the internet we have a different sort of
> opportunity. Let me try to describe it.
>
>    - What is missing in most mathematical presentations is a view into the
>    often very human and very messy process of getting to the presentation. What
>    young mathematicians need -- in my view -- is a view of mathematicians doing
>    mathematics. They need to see very top-down orientations rubbing elbows with
>    very bottoms-up orientations. They need to see highly inventive, unifying
>    viewpoints come up against skeptical viewpoints armed with vast arrays of
>    counter-examples. They need to see people desperately trying to organize
>    while others are desperately trying to de-construct. This is where the life
>    of mathematics is. This is how people bring mathematics to life.
>    - With the internet we have the opportunity to record not just the final
>    artifact, tractate or wiki, but the process. Ever since Andre Joyal
>    mentioned a 2nd life for Bourbaki i can't stop thinking about a Bourbaki
>    colloquium run in Second Life <http://secondlife.com/> -- so that
>    whatever the outcome of a given process is in terms of artifact, people can
>    go back and look at the process, itself. They can see how people argued and
>    counter-argued. There is getting to be a precendent for this, from
>    Harvard<http://www.joystiq.com/2006/09/12/harvard-class-invades-second-life/>to
>    Intel <http://softwarecommunity.intel.com/articles/eng/1283.htm>, to run
>    serious technical conversation in Second Life.
>
> Perhaps this idea is too far out, but i would urge those who seriously
> consider a second life for Bourbaki to remember to record the living part as
> well as the outcome. After all, looking over the last many emails to
> categories so much of it is an attempt to recover process -- how things got
> to be where they are.
>
> Best wishes,
>
> --greg
>

Re: Bourbaki and Categories

[Jacques Carette]

I hereby volunteer my time and the technical resources I have at my
disposal to 'host' a "New Bourbaki based on CT" project.

Why me?
1. I am an interested _user_ of category theory, but not an 'inventor'
in the field.  Neutrality is frequently an asset in a "keeper of the
infrastructure".
2. I need the results of such a project!  My core work (for 11 years in
industry and now 6 in academia) is in "mechanized mathematics", or
computer-based tools that help automate the mathematics process,
explicitly including both computation (i.e. what Maple does) and proof
(i.e. what Coq does).  (The curious can see
http://imps.mcmaster.ca/mathscheme/publications.html for some of our
work and my personal publications page for more).  One cannot
simultaneously build a library of mathematics and know category theory
without seeing its tendrils everywhere.
3. I have a real stake in seeing "mathematics on computers": not only is
it my day-to-day research, I am also the Chair of the Electronic
Services Committee for the Canadian Mathematical Society (CMS).  I am
actively searching for *good* applications of "electronic services"
where we can get involved.

I cannot promise the resources of the CMS for this, but I can certainly
promise to bring this up as an agenda item for our December meeting in
Ottawa.  However, I would be extremely proud if I could claim, years
from now, that "Bourwiki" was a ``truly international project which
benefited from a strong involvement of the Canadian Mathematical Society''.

Jacques Carette

PS: Below here are some minor comments/thoughts on the various topics in
this email thread -- the only important parts are above.
1. I am fairly fond of the (awfully titled) book "Post-modern Algebra"

http://www.amazon.com/Post-Modern-Algebra-Applied-Mathematics-Wiley-Interscience/dp/0471127388
  by J. Smith and A. Romanwska, as a middle-ground between full-fledged
use of categories and classical algebra in an introductory text.  The
organization of the text is definitely non-classical and tries to
organize concepts according to mathematical criteria rather than
historical timelines.
2. I definitely believe that a proper encoding of modern mathematics
into a 'library' should be done top-done by specializing abstract
concepts.  The "Little Theories" method from theorem proving is
essentially the recognition that a large body of mathematics is a huge
diagram in an appropriate category of theories. These issues are all too
frequently treated as meta-mathematical.  The constructive theory of
"representations of theories" (via sketches or otherwise) can play a
very important role in software construction.
3. Heuristics and examples are crucial for human understanding of
mathematics.  Thus, while a "mechanized mathematics system" may be built
from highly abstract pieces, it needs to present itself to its users in
as concrete a manner as possible.  This dichotomy between
system-building and usability is further explored in a paper "High-Level
Theories" (available officially at
http://www.springerlink.com/content/b1122523vtm88w73/, unofficially at
http://imps.mcmaster.ca/doc/hlt.pdf )
4. I cannot over-emphasize my agreement with Joyal's statement that "But
mathematics is naturally organised and simple!"
5. To a certain degree, some libraries (like those of the large theorem
provers) attempt "partially unified presentations of mathematics"; some
even have pseudo-databases.  These are unfortunately quite classical in
their structure, with the exception of the work of the Kestrel Institute
(in computer science) which is very categorical.  The algebra library of
the programming language 'Aldor' (www.aldor.org) was definitely
influenced by categories.  Category theory is actively influencing the
development of the language 'Haskell'.
6. What I propose to help with are the technical and online aspects of
Andre Joyal's proposal:
> A special database for mathematics should be created
> (but I dont know how).
> Papers in the NB section of the arXves could be selected,
> modified and organised with a system of references,
> to give a partially unified presentation of mathematics.
> The same database could support different
> presentations realised by different competing teams.
> Each team could work like a mathematical journal,
> with an editor in chief and an editorial board.
>
using means like the ones which Andrej Bauer suggested.

Bourbaki and Categories

[John Baez]

On Fri, Sep 19, 2008 at 10:16:31PM -0400, jim stasheff wrote:

> John Baez wrote:

> >I can imagine a new Bourbaki who tries to explain all of
> >mathematics in the language of categories.  But I can
> >also imagine a new Bourbaki who tries to explain all of
> >mathematics in the language of infinity-categories.

> Is it necessary to have a global point of view to appreciate Bourbaki? I
> found them quite valuable locally - i.e.. just a few of the chapters by
> themselves earned my appreciation.

It's easy to appreciate their books locally - but I think they
sought a global systematic viewpoint while writing them.

It's possible that a "neo-Bourbaki" should take a less systematic
approach.  Mathematics may be too much in a state of foundational
flux for a systematic approach to be successful right now.  Maybe
the best we can hope for is something a bit more like Wikipedia,
where different people contribute different portions of text, and
they don't cohere in a polished whole.  But presumably anyone calling
for a new Bourbaki wants something different from Wikipedia.

There's "Scholarpedia":

http://www.scholarpedia.org/

but it doesn't seem to be doing anything with math yet, and if
it ever does, I bet it'll take a "midde-of-the-road" approach instead
of pushing a specific intellectual agenda.

I would like to see lots of people try lots of different things.

Best,
jb

Re: Bourbaki and Categories

[Meredith Gregory]

All,

i have been utterly delighted by this conversation. What i can't help but
think about, however, is that with the internet we have a different sort of
opportunity. Let me try to describe it.

   - What is missing in most mathematical presentations is a view into the
   often very human and very messy process of getting to the presentation. What
   young mathematicians need -- in my view -- is a view of mathematicians doing
   mathematics. They need to see very top-down orientations rubbing elbows with
   very bottoms-up orientations. They need to see highly inventive, unifying
   viewpoints come up against skeptical viewpoints armed with vast arrays of
   counter-examples. They need to see people desperately trying to organize
   while others are desperately trying to de-construct. This is where the life
   of mathematics is. This is how people bring mathematics to life.
   - With the internet we have the opportunity to record not just the final
   artifact, tractate or wiki, but the process. Ever since Andre Joyal
   mentioned a 2nd life for Bourbaki i can't stop thinking about a Bourbaki
   colloquium run in Second Life <http://secondlife.com/> -- so that
   whatever the outcome of a given process is in terms of artifact, people can
   go back and look at the process, itself. They can see how people argued and
   counter-argued. There is getting to be a precendent for this, from
   Harvard<http://www.joystiq.com/2006/09/12/harvard-class-invades-second-life/>to
   Intel <http://softwarecommunity.intel.com/articles/eng/1283.htm>, to run
   serious technical conversation in Second Life.

Perhaps this idea is too far out, but i would urge those who seriously
consider a second life for Bourbaki to remember to record the living part as
well as the outcome. After all, looking over the last many emails to
categories so much of it is an attempt to recover process -- how things got
to be where they are.

Best wishes,

--greg

On Sat, Sep 20, 2008 at 1:21 PM, Andre Joyal <joyal.andre@uqam.ca> wrote:

> On Fri, Sep 19, 2008, John Baez wrote:
>
> >Do we imagine this new Bourbaki as just systematizing and
> >presenting what we know already, or struggling to create
> >brand new mathematics?
>
> This is  an important question.
> We need to have a clear view of the goal of such an enterprise.
> From my point of view, the goal should be "educational":
> to help students and researchers to learn mathematics
> and cross the boundary between fields.
> Mathematics is vast, and every mathematician is a permanent student.
> The traditional way to learn is to read the litterature and to discuss with
> a master.
> I was told that Grothendieck had learned algebraic geometry by discussing
> with Serre.
> But few peoples have this chance.
> Obviously, Internet is opening new avenues for learning.
> Many peoples (and myself) have learned a lot by reading your bulletin
> "This Week's Finds in Mathematical Physics".
> You have a real talent to explain a subject by exposing the heuristic!
> A discussion forum like the "Categories list" is also very helpful.
> Wikipedia is a useful place to gather informations about a subject.
> But the Bourbaki Tractate was offering something more:
> a unified presentation of mathematics, including the proofs.
>
> The Bourbaki Tractate was the result of a sustained collaboration
> of many generations of mathematicians from different fields.
> Conflicts are inevitable and mathematics evolve quickly.
> A unified, final presentation seems impossible.
>

Bourbaki and Categories

[Andre Joyal]

On Fri, Sep 19, 2008, John Baez wrote:

>Do we imagine this new Bourbaki as just systematizing and
>presenting what we know already, or struggling to create
>brand new mathematics?

This is  an important question.
We need to have a clear view of the goal of such an enterprise. 
>From my point of view, the goal should be "educational":
to help students and researchers to learn mathematics
and cross the boundary between fields. 
Mathematics is vast, and every mathematician is a permanent student. 
The traditional way to learn is to read the litterature and to discuss with a master.
I was told that Grothendieck had learned algebraic geometry by discussing with Serre.
But few peoples have this chance.
Obviously, Internet is opening new avenues for learning. 
Many peoples (and myself) have learned a lot by reading your bulletin 
"This Week's Finds in Mathematical Physics". 
You have a real talent to explain a subject by exposing the heuristic!
A discussion forum like the "Categories list" is also very helpful.
Wikipedia is a useful place to gather informations about a subject.
But the Bourbaki Tractate was offering something more:
a unified presentation of mathematics, including the proofs.

The Bourbaki Tractate was the result of a sustained collaboration 
of many generations of mathematicians from different fields. 
Conflicts are inevitable and mathematics evolve quickly.
A unified, final presentation seems impossible.

On Thur, Sep 18, 2008, Ronnie Brown wrote: 

> Something initially less ambitious like this
>might actually get done. Being electronic, it would be  seen as a `current',
>rather than `final account', and so would better reflect the way mathematics
>develops, in which a slight shift of emphasis, or notation (like --> for a
>function), can have profound consequences.

A partially unified evolving presentation of mathematics seems possible.
Today's littérature is already offering something like that!
But I find it cahotic and complicated.
But mathematics is naturally organised and simple!
The complexity of the litterature is often artificial.
Many proofs are complicated, simply because the author
ignores the abstract argument that could simplify everything.
Some statements are left unproved, and peremptory declared obvious
when they are not. The reader who cant see the obvious thing is terrorised.
He may as well quit mathematics. 

What can we do?

Let me submit a few ideas for disccusion.

Mathematics is naturally self-organised. 
The proof of most theorems can be broken in small steps of the form 
A_1,..,A_n --->B, where A_1,..,A_n is the list
of hypothesis and B is the conclusion. 
Each step may have a simple proof.
A complete proof maybe obtained by working
backward from the statement of the theorem to 
the axioms ot to known theorems. 

Everyone who knows a nice proof of a meaningful 
implication A_1,..,A_n --->B should write
a paper about it and put it in a special section of the arXiv (the NB section ?).

On Mon, Sep 19, 2008, Michael Spivack wrote:

>Even more interesting to me would be a kind of zoom-feature on 
>proofs.  Proofs are in the eye of the beholder: for example it has 
>been debated as to whether Perelman's 70 pages was a full proof of 
>geometrization.  Given a proof with a statement which one does not 
>understand, a mathematician may find himself reproving something that 
>was obvious to (or wrongly assumed to be obvious by) another 
>mathematician.  The community could benefit if a mathematician who 
>proves such a statement then uploaded the proof, even in rough form, 
>to some kind of math wiki.  If it were well-organized, this math wiki 
>could revolutionize how mathematics is done.  In fact, choosing the 
>"right way" to organize such a site may itself be a problem which 
>could produce interesting mathematics.

On Tues, Sep 16, 2008, Andrej Bauer wrote:

>My opinion is that we have not yet found the right way to do
>"hive-science", but when we do, it will be a revolution. (A good start
>would be to get out of the hold that the evil publishers have on us.)

A special database for mathematics should be created
(but I dont know how).
Papers in the NB section of the arXves could be selected,
modified and organised with a system of references,
to give a partially unified presentation of mathematics. 
The same database could support different
presentations realised by different competing teams.  
Each team could work like a mathematical journal,
with an editor in chief and an editorial board.

What do you think?

Andre

Re: Bourbaki and Categories

[Zinovy Diskin]

Let me add my two cents :)

On Tue, Sep 16, 2008 at 4:57 AM, Vaughan Pratt <pratt@cs.stanford.edu> wrote:

> If the subject Bourwiki is proposing to serve is mathematics, then
> perhaps it is time that the American Mathematical Monthly, along with
> the Putnam Mathematical Competition, the International Mathematics
> Olympiad, and the Journal of the AMS, abandon their pretense of being
> about mathematics and come up with a suitable name for their subject.

We can think of two definitions of math. The first is based on the
subject matter ("what"): math is the study of Bourbaki's structures.
The other is based on "how:"  math is the study of structures in a
well-structured way. With this definition, a good computer programmer
or, say, Henry Ford, who applied conveyor to assembling, are more
mathematicians than some of the guys responsible for what Vaughan
wrote about.

If we imagine a two dimensional plane with the "what" axis being
vertical and the "how" horizontal, then we get two mathematics and
resp. two sorts of mathematicians: vertical and horizontal. (CT and
CT-rists go, of course, along the harmonized diagonal :).
Historically, Bourbaki put a bold point on the line of
Euclid-Peano-Hilbert and indeed formed the vertical dimension of the
modern mathematical space.  However, while the very texts written by
Bourbaki are mostly enjoyable, his epigones have created a special
literary style, which is good for writing/producing but hardly for
reading mathematical papers.  Bourbaki should not be blamed for wide
dissemination of this indigestible style but...Vladimir Arnold once
said that  "bourbakization" of modern mathematics should perhaps be
called "oBourbachivanie" in Russian (which rhymes with the Russian
"oDourachivanie", which refers to fooling with someone :).

> Not only do categories, functors, natural transformations, adjunctions,
> and monads go unused in these 20th century icons of mathematics, they go
> unacknowledged.  Clearly they have not gotten with the modern
> mathematical program and fall somewhere between a throwback to a golden
> age and a backwater of mathematics.  When they die off like the
> dinosaurs they are, real mathematics will be able to advance unfettered
> into the 21st century and beyond.
>
Dinosaurs would not normally die off themselves. Some causes are
needed, and here's one  (somewhat speculative though). CT can change
the very notion of what a formal definition is. In the modern style,
the notion of ordered pair/tuple and its derivatives like formula and
term are central.  This quite simple syntax (as is often happens with
simple syntax, think, for example, of a Java program)  can hide
complex structures so that a tuple-based formal definition is not
actually formal and implicitly involves intuitive concepts. If CT will
sometime indeed reshape the criteria of being a formal specification
(of a Bourbaki's structure), then dinosaurs would be forced to acquire
CT (or die off).

Zinovy


> Judging from the talks at BLAST in Denver last month (B = Boolean
> algebras, L = lattices, A = (universal) algebra, S = set theory, T =
> topology), at least the algebraic community is moving very slightly in
> this direction.  Things will hopefully improve yet further when
> algebraic geometry gets over its snit with equational model theory.
>
> Meanwhile if you need a witness for seven degrees of separation, look no
> further than AMM and CT.
>
> (I confess to being an unreconstructed graph theorist and algebraist
> myself.  I may have to preemptively volunteer myself for re-education
> before it becomes involuntary.)
>
> Vaughan Pratt
>
>
>

Re: Bourbaki and Categories

[jim stasheff]

John Baez wrote:
> On Wed, Sep 17, 2008 at 01:13:21PM -0400, Andre Joyal wrote:
>
>
>> Is it the time for a new Bourbaki?
>>
>
> Do we imagine this new Bourbaki as just systematizing and
> presenting what we know already, or struggling to create
> brand new mathematics?
>
> I can imagine a new Bourbaki who tries to explain all of
> mathematics in the language of categories.  But I can
> also imagine a new Bourbaki who tries to explain all of
> mathematics in the language of infinity-categories.  It
> may be a bit too late for the first one, and a bit too
> early for the second one.
>
> Perhaps we need both!
>
> Best,
> jb
>
>
>
>
>
Is it necessary to have a global point of view to appreciate Bourbaki? I
found them quite valuable locally - i.e.. just afew of the chapters by
themselves
earned my appreciaiton.

jim

Re: Bourbaki and Categories

[Mark.Weber]

Dear Michael,

Semantically, as Lawvere observed long ago, a monad gives rise not just to
a category of algebras but also to a forgetful functor into the category
on which the monad acts. For any category C the functor

"semantics" : Mnd(C)^op --> CAT/C

whose object map sends a monad on C to its associated forgetful functor is
full and faithful. Thus a pair of monads on C giving rise to isomorphic
forgetful functors must necessarily be isomorphic. So your observations
about different monads giving rise to the same algebras, while correct, do
not tell the whole story on the semantic side.

The situation is of course different for sketches: they too give rise to
forgetful functors (into Set), but this does not suffice to determine a
given sketch up to isomorphism in the same way, and this justifies Steve
Lack's perspective of "sketches as presentations of theories".

Mark Weber

> Previously, Michael Barr wrote:
>
> Of course sketches are mathematical objects in their own right.  Of
> course, the functor that assigns to each sketch the corresponding theory
> is not full or faithful.  But the definition is precise, the notion of
> model is also precise, so I have no idea what, if any, content there is in
> the claim.  Incidentally, you might with equal justice claim that triples
> are not mathematical objects since two distinct triples can have
> isomorphic categories of Eilenberg-Moore algebras.  In fact there are
> triples (or theories) on Set that have infinitary operations, yet whose
> category of models is isomorphic to Set.
>
> Michael
>

Re: Bourbaki and Categories

[Zinovy Diskin]

Dear Michael,

Still some content in Steve's claim could be imagined. A working
mathematician (WM) works with Borubaki's structures like groups or
vector spaces and leaves all worries about what his proofs actually
mean for a working math logician.  For such a WM, sketches (as any
other syntactical machineries) are indeed technical minutiae rather
than mathematical objects. It'd be perhaps a reasonable view unless a
bunch of strong semantic results (Tarski, Mal'cev,Robinson) that our
WM values so much, which are provided by bringing syntax onto the
stage.

Zinovy

On Thu, Sep 18, 2008 at 10:31 AM, Michael Barr <barr@math.mcgill.ca> wrote:
> Of course sketches are mathematical objects in their own right.  Of
> course, the functor that assigns to each sketch the corresponding theory
> is not full or faithful.  But the definition is precise, the notion of
> model is also precise, so I have no idea what, if any, content there is in
> the claim.  Incidentally, you might with equal justice claim that triples
> are not mathematical objects since two distinct triples can have
> isomorphic categories of Eilenberg-Moore algebras.  In fact there are
> triples (or theories) on Set that have infinitary operations, yet whose
> category of models is isomorphic to Set.
>
> Michael
>

Bourbaki and Categories

[John Baez]

On Wed, Sep 17, 2008 at 01:13:21PM -0400, Andre Joyal wrote:

> Is it the time for a new Bourbaki?

Do we imagine this new Bourbaki as just systematizing and
presenting what we know already, or struggling to create
brand new mathematics?

I can imagine a new Bourbaki who tries to explain all of
mathematics in the language of categories.  But I can
also imagine a new Bourbaki who tries to explain all of
mathematics in the language of infinity-categories.  It
may be a bit too late for the first one, and a bit too
early for the second one.

Perhaps we need both!

Best,
jb

Re: Bourbaki and Categories

[Andree Ehresmann]

George Janedlize writes

> Could you please explain this better...
> the Bourbaki group simply did not see the importance of category theory...
> However, there were three great category-theorists in that group...
> I have never heard of any joint work of Charles Ehresmann  with any  
> of the two others, Eilenberg and Grothendieck... ...the relation  
> between Bourbaki Tractate and category theory should have been  
> determined by their separate or joint influence and therefore also  
> by their communication with each other (if any).

I'll try to explain why there is no contradiction.
1. Charles only participated actively to the Bourbaki group from 1935  
to the mid forties, at a time he did not know category theory. In 1935  
he had written a first version for the volume "Theorie des ensembles"  
where he introduced the notions of local structures and associated  
pseudogroups of transformations (not so far from groupoids!), but this  
version was not accepted and he did not like the final version  
published much later. After the war, he only participated irregularly  
because he felt that he was no more able to make himself heard, the  
decisions being taken by "those who spoke the more loudly" (as he said  
to me).
2. Around 1950 it was decided that active participation ended at 45  
(the age Charles had then), lessening the influence of those  
(Eilenberg, Cartan, Chevalley and Dieudonne) who  could have stressed  
the importance of categories. I don't know exactly when Grothendieck  
became a member, but it was much later, and I think he did not remain  
for long. Later on, disdain for category theory had developed in  
France...
3. As for the communication between Charles and the other  
category-theorists, he had no contact with Grothendieck who was much  
younger. He was friendly with Eilenberg but did not see him often.  
Before the war he lived in Paris and regularly met Henri Cartan,  
Dieudonne, and more specially, Chevalley (both had regular exchanges  
with the philosophers Cavailles and Lautman). But their communication  
almost ceased after the war when he developed all his activity in  
Strasbourg (up to 1955) and was out of France for a long part of the  
year. Anyway, before our joint work (from the mid sixties up to his  
death), Charles worked essentially alone and published no joint work  
at all, except 6 Notes on Topology or Geometry with some of his  
students. When he began to specialize in category theory in the  
sixties, it was not well understood by other French mathematicians,  
and his influence dwindled up to a real opposition in the seventies.

Andree

Re: Bourbaki and Categories

[cat-dist]

Michael,

That seems a very good and generous idea!

It could  initially be posted on your site, except that it should also be
indefinitely available.

The pdf should have hyperref, and perhaps there should be a complete version
and also a parcelled version, so that wiki and Planet Math could have links
to specific parts. What do you think?

Ronnie




----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "R Brown" <ronnie.profbrown@btinternet.com>
Cc: "Vaughan Pratt" <pratt@cs.stanford.edu>; <categories@mta.ca>
Sent: Thursday, September 18, 2008 3:36 PM
Subject: Re: categories: Re: Bourbaki and Categories


> The three books I have authored or co-authored all contain a chapter of
> about 40 pages that is an introduction to category theory.  They are
> largely identical (the one for Acyclic Models has a an added section on
> categories of fractions, needed for that book).  The one in TTT is already
> freely available and, with Charles's permission, I would happily post that
> in whatever place you would like.
>
> Michael
>

Re: Bourbaki and Categories

[Michael Barr]

The three books I have authored or co-authored all contain a chapter of
about 40 pages that is an introduction to category theory.  They are
largely identical (the one for Acyclic Models has a an added section on
categories of fractions, needed for that book).  The one in TTT is already
freely available and, with Charles's permission, I would happily post that
in whatever place you would like.

Michael

On Wed, 17 Sep 2008, R Brown wrote:

> There are already some pretty good categorical entries on wiki; I have
> modified some of the entries on groups, actions, equivalence relations,  to
> include references to groupoids, which has resulted in hits. But we should
> also consider planetmath.org (entries are contributed under the terms of the
> GNU Free Documentation License (FDL)) which allows for group work and is not
> so open as wiki to general modification. It needs a group of you wonderful
> energetic people to engage with reviewing what is on wiki and planetmath and
> making sure they express what is fealt to be desirable!
>

...

Re: Bourbaki and Categories

[Michael Barr]

Of course sketches are mathematical objects in their own right.  Of
course, the functor that assigns to each sketch the corresponding theory
is not full or faithful.  But the definition is precise, the notion of
model is also precise, so I have no idea what, if any, content there is in
the claim.  Incidentally, you might with equal justice claim that triples
are not mathematical objects since two distinct triples can have
isomorphic categories of Eilenberg-Moore algebras.  In fact there are
triples (or theories) on Set that have infinitary operations, yet whose
category of models is isomorphic to Set.

Michael

On Wed, 17 Sep 2008, Steve Lack wrote:

> Dear Andrei,
>
> Sketches are not mathematical objects in their own right, in the same sense
> that groups or spaces are. They are presentations (for theories), and have
> status similar to other sorts of presentations (for groups, rings, etc.)
>
> Of course that is in no way meant to suggest that they are not important and
> worthy of study.
>
> Regards,
>
> Steve Lack.
>
>
> On 16/09/08 11:09 PM, "Andre.Rodin@ens.fr" <Andre.Rodin@ens.fr> wrote:
>
>>
>> Of course, you are right about a point, I missed it! I must confess I didn't
>> think about this example in precise terms. My claim is that sketch theory
>> doesn't fit the structuralist (Bourbaki-Hilbertian) pattern. It hardly
>> precisely fits the ancient Euclidean pattern either but there is a suggestive
>> analogy, which concerns the idea that certain basic objects like point, line
>> and circle *generate* the rest.
>> A further claim is this: a specific reason *why* sketch theory doesn't fit the
>> structuralist pattern is that in sketch theory (like in CT in general)
>> isomorphisms don't have the same distinguished status.
>>
>> andrei
>>
>>
>>
>>
>>
>>> I don't know what to say about the suggestion that a circle and a line
>>> make a sketch of which Euclidean plane geometry is a model.  I would think
>>> you would need a point too, since intersections are crucial.  Maybe
>>> complex projective geometry since then two lines intersect in one point
>>> (unless they coincide), a line and a circle in two (unless they are
>>> tangent or equal) and every pair of circles in four (ditto).  Maybe the
>>> exceptions could be handled in some sketch.  At any rate, it wold e
>>> interesting to try to sketch this in detail.  At any rate, I never thought
>>> about this before.
>>
>>> Michael
>>
>>
>>
>
>
>

Re: Bourbaki and Categories

[Andre Joyal]

Dear George,

I thank you for your message.

You wrote:

>I insist that Bourbaki group simply did
>not see the importance of category theory.

It is difficult to know. The Boubaki group had shielded itself
in secrecy, like a free mason cell. 
You are surely aware of the interview of Pierre Cartier 
in the Mathematical Intelligencer No1 1998. 
Everyone interested in the history of Bourbaki should read it.

http://ega-math.narod.ru/Bbaki/Cartier.htm

Let me stress a few passages: 

>The fourth generation was more or less a group of students of Grothendieck. But at that time Grothendieck had already left Bourbaki. 
>He belonged to Bourbaki for about ten years but he left in anger. The personalities were very strong at the time. 
>I remember there were clashes very often. There was also, as usual, a fight of generations, like in any family. 
>I think a small group like that repeated more or less the psychological features of a family. 
>So we had clashes between generations, clashes between brothers, and so on. 
>But they did not distract Bourbaki from his main goal, even though they were quite brutal occasionally. 
>At least the goal was clear. There were a few people who could not take the burden of this psychological style, 
>for instance Grothendieck left and also Lang dropped out.


> It is amazing that category theory was more or less the brainchild of Bourbaki. The two founders were Eilenberg and MacLane. 
> MacLane was never a member of Bourbaki, but Eilenberg was, and MacLane was close in spirit. The first textbook on homo-logical 
>algebra was Cartan-Eilenberg, which was published when both were very active in Bourbaki. Let us also mention Grothendieck, who 
>developed categories to a very large extent. I have been using categories in a conscious or unconscious way in much of my work, 
>and so had most of the Bourbaki members. But because the way of thinking was too dogmatic, or at least the presentation in the 
>books was too dogmatic, Bourbaki could not accommodate a change of emphasis, once the publication process was started.

>In accordance with Hilbert's views, set theory was thought by Bourbaki to provide that badly needed general framework. If you need 
> some logical foundations, categories are a more flexible tool than set theory. The point is that categories offer both a general 
> philosophical foundation—that is the encyclopedic, or taxonomic part—and a very efficient mathematical tool, to be used in 
>mathematical situations. That set theory and structures are, by contrast, more rigid can be seen by reading the final chapter in 
>Bourbaki set theory, with a monstrous endeavor to formulate categories without categories.


The interview ends with the following passage.
The bold faces are from me.

>When I began in mathematics the main task of a mathematician was to bring order and make a synthesis of existing material, to create 
>what Thomas Kuhn called normal science. Mathematics, in the forties and fifties, was undergoing what Kuhn calls a solidification period. 
>In a given science there are times when you have to take all the existing material and create a unified terminology, unified standards, 
>and train people in a unified style. The purpose of mathematics, in the fifties and sixties, was that, to create a new era of normal science. 
>Now we are again at the beginning of a new revolution. Mathematics is undergoing major changes. We don't know exactly where it will go.
> It is not yet time to make a synthesis of all these things—MAYBE IN TWENTY OR THIRTY CENTURY IT WILL BE TIME FOR A NEW BOURBAKI.
>I consider myself very fortunate to have had two lives, a life of normal science and a life of scientific revolution.

Is it the time for a new Bourbaki?

Best regards, 
André


-------- Message d'origine--------
De: cat-dist@mta.ca de la part de George Janelidze
Date: lun. 15/09/2008 20:03
À: categories@mta.ca
Objet : categories: Re:  Bourbaki and Categories
 
Dear Andree,

Could you please explain this better?:

The only Bourbaki member I new personally was Sammy Eilenberg. As many of
us, I knew him very well and I would say that he was more skeptical about
the Bourbaki Tractate then one can conclude from Andre's message. Having in
mind not just this but the content of Bourbaki's "Homological algebra" and
what we see today from the followers of that Bourbaki group, I protest
against Andre's "two options" and I insist that Bourbaki group simply did
not see the importance of category theory (in spite of being brilliant
mathematicians, as I said in my previous message). I hope Andre will forgive
me and even agree with me.

However, there were three great category-theorists in that group (plus there
is this mysterious story about Chevalley's book of category theory lost in
the train), and "did not see" cannot be said about them of course. On the
other hand I have never heard of any joint work of Charles Ehresmann with
any of the two others, Eilenberg and Grothendieck (and nothing jointly from
them). I think apart from the time issues you describe, the relationship
between Bourbaki Tractate and category theory should have been determined by
their separate or joint influence and therefore also by their communication
with each other (if any).

Is this true, and could you please give details?

Respectfully, and with best regards-

George

Re: Bourbaki and Categories

[R Brown]

There are already some pretty good categorical entries on wiki; I have
modified some of the entries on groups, actions, equivalence relations,  to
include references to groupoids, which has resulted in hits. But we should
also consider planetmath.org (entries are contributed under the terms of the
GNU Free Documentation License (FDL)) which allows for group work and is not
so open as wiki to general modification. It needs a group of you wonderful
energetic people to engage with reviewing what is on wiki and planetmath and
making sure they express what is fealt to be desirable!

In the old days, a graduate book would have an appendix on say set theory,
and maybe basic algebra, as needed for the rest of the text. It would be
very useful to have basic category theory (in terms of `the basic facts of
life') on the web available to all, with nice accounts of say `left adjoints
preserve colimits', etc. , with many convincing examples, and maybe history,
to which a text could refer. Something initially less ambitious like this
might actually get done. Being electronic, it would be  seen as a `current',
rather than `final account', and so would better reflect the way mathematics
develops, in which a slight shift of emphasis, or notation (like --> for a
function), can have profound consequences.

There is perhaps a  case for a separate collected electronic account, with
hyperref, and also a printed version, since a book is a useful portable
random access device. Print on Demand allows this to be produced quite
cheaply, with a 35% royalty on retail sales, and available on amazon.

Ronnie

----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: <categories@mta.ca>
Sent: Tuesday, September 16, 2008 9:57 AM
Subject: categories: Re: Bourbaki and Categories


> Bourbaki redone as Bourwiki (thanks, Dusko!) with the benefit of
> category theoretic insights will hopefully clarify some segments of
> mathematics.
>
> What troubles me in this discussion however is its assumed scope of
> "some."  I get the sense that there are people who want it to be
> mandated as "all."
>
> Perhaps it should be.
>
> Just now I looked through an issue of American Mathematical Monthly that
> came to hand to get a sense of the likely alignment of Bourwiki with
> what the mathematical community generally regards as the scope of its
> subject.  Actually I do this periodically, and I don't see much change
> between the issue I picked up just now and any of the other issues I've
> looked at in the past with just this question in mind.
>
> If the subject Bourwiki is proposing to serve is mathematics, then
> perhaps it is time that the American Mathematical Monthly, along with
> the Putnam Mathematical Competition, the International Mathematics
> Olympiad, and the Journal of the AMS, abandon their pretense of being
> about mathematics and come up with a suitable name for their subject.
> Not only do categories, functors, natural transformations, adjunctions,
> and monads go unused in these 20th century icons of mathematics, they go
> unacknowledged.  Clearly they have not gotten with the modern
> mathematical program and fall somewhere between a throwback to a golden
> age and a backwater of mathematics.  When they die off like the
> dinosaurs they are, real mathematics will be able to advance unfettered
> into the 21st century and beyond.
>
> Judging from the talks at BLAST in Denver last month (B = Boolean
> algebras, L = lattices, A = (universal) algebra, S = set theory, T =
> topology), at least the algebraic community is moving very slightly in
> this direction.  Things will hopefully improve yet further when
> algebraic geometry gets over its snit with equational model theory.
>
> Meanwhile if you need a witness for seven degrees of separation, look no
> further than AMM and CT.
>
> (I confess to being an unreconstructed graph theorist and algebraist
> myself.  I may have to preemptively volunteer myself for re-education
> before it becomes involuntary.)
>
> Vaughan Pratt
>
>

Re: Bourbaki and Categories

[Andre.Rodin]

Dear Steve,


> Sketches are not mathematical objects in their own right, in the same sense
> that groups or spaces are.

Of course, they are not.


>They are presentations (for theories), and have
> status similar to other sorts of presentations (for groups, rings, etc.)


I think about a sketch as an alternative to a string of formulae, which
represents (axioms of) a theory. I didn't try to compare sketch theory with
group theory.  I tried to compare sketch theory with the general setting, in
which group theory is developed a la Bourbaki (along with many other
theories). A classical account of this general setting (which differs at certain
points with Bourbaki's version) is Tarski's model theory.

The notion of presentation in my understanding implies that what a given
presentation is a presentation *of* is somehow given in advance. I try to think
of a sketch as a means to build a theory, not to present a ready-made theory.
Perhaps *representation* is a better word for it than *presentation*.

best,
andrei

Re: Bourbaki and Categories

[Andre.Rodin]

Certainly! I shall try. Thank you for encouraging! This issue seems me also
interesting from a different viewpoint. Even if the New Maths was a pedagogical
failure it is still remarkable that the traditional school mathematics can be
wholly spelled out in the Bourbaki-like terms. I wonder if CT allows for
anything of this sort.

andrei



>Still, it might b interesting and even instructive to try to build a
>sketch whose objects are (interpreted as) sets of points sets of lines,
>sets of circles and intersection is an operation.  I guess incidence would
>have to a relation.  It might work.

>Michael

Re: Bourbaki and Categories

[Michael Barr]

Still, it might be interesting and even instructive to try to build a
sketch whose objects are (interpreted as) sets of points sets of lines,
sets of circles and intersection is an operation.  I guess incidence would
have to be a relation.  It might work.

Michael

Re: Bourbaki and Categories

[jim stasheff]

David Spivak wrote:
> I agree with Andre.  Encapsulating a group of mathematicians inside a
> single named entity fosters a kind of collaborative spirit in which
> good ideas are not kept for personal use later but are shared amongst
> the community.  When ideas are shared in real time, good mathematics
> can be produced faster.  Anyone who wants to join the collective can
> do so, and the collective produces highly useful material.  Of course
> such an enterprise is orthogonal to name-recognition, and maybe to
> getting tenure!

A partial solution: those who already have sufficient name recognition
should proclaim the value contributed by those without it and especially
in support of tenure for them.

jim

Re: Bourbaki and Categories

[Andre.Rodin]

Of course, you are right about a point, I missed it! I must confess I didn't
think about this example in precise terms. My claim is that sketch theory
doesn't fit the structuralist (Bourbaki-Hilbertian) pattern. It hardly
precisely fits the ancient Euclidean pattern either but there is a suggestive
analogy, which concerns the idea that certain basic objects like point, line
and circle *generate* the rest.
A further claim is this: a specific reason *why* sketch theory doesn't fit the
structuralist pattern is that in sketch theory (like in CT in general)
isomorphisms don't have the same distinguished status.

andrei





>I don't know what to say about the suggestion that a circle and a line
>make a sketch of which Euclidean plane geometry is a model.  I would think
>you would need a point too, since intersections are crucial.  Maybe
>complex projective geometry since then two lines intersect in one point
>(unless they coincide), a line and a circle in two (unless they are
>tangent or equal) and every pair of circles in four (ditto).  Maybe the
>exceptions could be handled in some sketch.  At any rate, it wold e
>interesting to try to sketch this in detail.  At any rate, I never thought
>about this before.

>Michael

Re: Bourbaki and Categories

[Michael Barr]

I don't know what to say about the suggestion that a circle and a line 
make a sketch of which Euclidean plane geometry is a model.  I would think 
you would need a point too, since intersections are crucial.  Maybe 
complex projective geometry since then two lines intersect in one point 
(unless they coincide), a line and a circle in two (unless they are 
tangent or equal) and every pair of circles in four (ditto).  Maybe the 
exceptions could be handled in some sketch.  At any rate, it wold e 
interesting to try to sketch this in detail.  At any rate, I never thought 
about this before.

Michael

On Tue, 16 Sep 2008, Andre.Rodin@ens.fr wrote:

>
> When one defines, say, a group à la Borbaki, i.e. structurally, it usually goes
> without saying that the defined structure is defined up to isomorphism. The
> notion of isomorphism plays in this case the role similar to that of equality
> in the (naive) arithmetic. In most structural constexts the distinction between
> the "same" structure and isomorphic structures is mathematically trivial just
> like the distinction between the "same" number and equal numbers. It may be not
> specially discussed in this case exactly because it is very basic. The notion
> of admissible map, say, that of group homomorphism, on the contrary, requires a
> definition, which may be non-trivial.
> The idea to do mathematics up to isomorphism is not Bourbaki's invention; it
> goes back at least to Hilbert's "Grundlagen der Geometrie" of 1899. In this
> sense the modern axiomatic method is structuralist. In his often-quoted letter
> to Frege Hilbert explicitely says that a theory is "merely a framework" while
> domains of their objects are multiple and transform into each other by
> "univocal and reversible one-one transformations". Those who trace the history
> of mathematical structuralism back to Hilbert are quite right, in my view.
> I have in mind two issues related to CT, which suggest that CT goes in a
> *different* direction - in spite of the fact that MacLane and many other
> workers in CT had (and still have) structuralist motivations. The first is
> Functorial Semantics, which brings a *category* of models, not just one model
> up to isomorphism. From the structuralist viewpoint the presence of
> non-isomorphic models (i.e. non-categoricity) is a shortcoming of a given
> theory. From the perspective of Functorial Semantics it is a "natural" feature
> of mathematical theories to be dealt with rather than to be remedied.
> The second thing I have in mind is Sketch theory. I cannot see that Hilbert's
> basic structuralist intuition applies in this case. In my understanding things
> work in Sketch theory more like in Euclid. Think about circle and straight line
> as a sketch of the theory of the first four books of Euclid's "Elements". I
> would particularly appreciate, Michael, your comment on this point since I
> learnt a lot of Sketch theory from your works.
> I have also a comment about the idea to rewrite Bourbaki's "Elements" from a new
> categorical viewpoint. Bourbaki took Euclid's "Elements" as a model for his
> work just like did Hilbert writing his "Gundlagen". In my view, this is this
> long-term Euclidean tradition of "working foundations", which is worth to be
> saved and further developed, in particular in a categorical setting. I'm less
> sure that Bourbaki's example should be followed in a more specific sense.
> Bourbaki tries to cover too much - and doesn't try to distinguish between what
> belongs to foundations and what doesn't. As a result the work is too long and
> not particularly usefull for (early) beginners. I realise that today's
> mathematics unlike mathematics of Euclid's time is vast, so it is more
> difficult to present its basics in a concentrated form. But consider Hilbert's
> "Grundlagen". It covers very little - actually near to nothing - of geometry of
> its time. But at the same time it provided a very powerful model of how to do
> mathematics in a new way, which greatly influenced mathematics education and
> mathematical research in 20th century. In my view, Euclid's "Elements" and
> Hilbert's "Grundlagen" are better examples to be followed.
>
> best,
> andrei
>
>
> le 15/09/08 12:59, Michael Barr à barr@math.mcgill.ca a écrit :
>
>> I don't know about this.  I took several courses in the late 1950s that
>> seem to have been influenced by the structuralist ideas (certainly
>> categories weren't mentioned; I never heard the word until Dave Harrison
>> arrived in 1959) and each of them started by defining an appropriate
>> notion of "admissible map".  I do not recall any special point being made
>> of isomorphism and I think in general it was used for what we now call a
>> bimorphism (1-1 and onto) even in cases, such as topological groups, when
>> they were not isomorphisms.
>>
>> To be sure Bourbaki was not mentioned either, but this structuralist
>> influence seemed strong.
>>
>> Michael
>>
>

Re: Bourbaki and Categories

[Andre.Rodin]

When one defines, say, a group à la Borbaki, i.e. structurally, it usually goes
without saying that the defined structure is defined up to isomorphism. The
notion of isomorphism plays in this case the role similar to that of equality
in the (naive) arithmetic. In most structural constexts the distinction between
the "same" structure and isomorphic structures is mathematically trivial just
like the distinction between the "same" number and equal numbers. It may be not
specially discussed in this case exactly because it is very basic. The notion
of admissible map, say, that of group homomorphism, on the contrary, requires a
definition, which may be non-trivial.
The idea to do mathematics up to isomorphism is not Bourbaki's invention; it
goes back at least to Hilbert's "Grundlagen der Geometrie" of 1899. In this
sense the modern axiomatic method is structuralist. In his often-quoted letter
to Frege Hilbert explicitely says that a theory is "merely a framework" while
domains of their objects are multiple and transform into each other by
"univocal and reversible one-one transformations". Those who trace the history
of mathematical structuralism back to Hilbert are quite right, in my view.
I have in mind two issues related to CT, which suggest that CT goes in a
*different* direction - in spite of the fact that MacLane and many other
workers in CT had (and still have) structuralist motivations. The first is
Functorial Semantics, which brings a *category* of models, not just one model
up to isomorphism. From the structuralist viewpoint the presence of
non-isomorphic models (i.e. non-categoricity) is a shortcoming of a given
theory. From the perspective of Functorial Semantics it is a "natural" feature
of mathematical theories to be dealt with rather than to be remedied.
The second thing I have in mind is Sketch theory. I cannot see that Hilbert's
basic structuralist intuition applies in this case. In my understanding things
work in Sketch theory more like in Euclid. Think about circle and straight line
as a sketch of the theory of the first four books of Euclid's "Elements". I
would particularly appreciate, Michael, your comment on this point since I
learnt a lot of Sketch theory from your works.
I have also a comment about the idea to rewrite Bourbaki's "Elements" from a new
categorical viewpoint. Bourbaki took Euclid's "Elements" as a model for his
work just like did Hilbert writing his "Gundlagen". In my view, this is this
long-term Euclidean tradition of "working foundations", which is worth to be
saved and further developed, in particular in a categorical setting. I'm less
sure that Bourbaki's example should be followed in a more specific sense.
Bourbaki tries to cover too much - and doesn't try to distinguish between what
belongs to foundations and what doesn't. As a result the work is too long and
not particularly usefull for (early) beginners. I realise that today's
mathematics unlike mathematics of Euclid's time is vast, so it is more
difficult to present its basics in a concentrated form. But consider Hilbert's
"Grundlagen". It covers very little - actually near to nothing - of geometry of
its time. But at the same time it provided a very powerful model of how to do
mathematics in a new way, which greatly influenced mathematics education and
mathematical research in 20th century. In my view, Euclid's "Elements" and
Hilbert's "Grundlagen" are better examples to be followed.

best,
andrei


le 15/09/08 12:59, Michael Barr à barr@math.mcgill.ca a écrit :

> I don't know about this.  I took several courses in the late 1950s that
> seem to have been influenced by the structuralist ideas (certainly
> categories weren't mentioned; I never heard the word until Dave Harrison
> arrived in 1959) and each of them started by defining an appropriate
> notion of "admissible map".  I do not recall any special point being made
> of isomorphism and I think in general it was used for what we now call a
> bimorphism (1-1 and onto) even in cases, such as topological groups, when
> they were not isomorphisms.
>
> To be sure Bourbaki was not mentioned either, but this structuralist
> influence seemed strong.
>
> Michael
>

Re: Bourbaki and Categories

[Vaughan Pratt]

Bourbaki redone as Bourwiki (thanks, Dusko!) with the benefit of
category theoretic insights will hopefully clarify some segments of
mathematics.

What troubles me in this discussion however is its assumed scope of
"some."  I get the sense that there are people who want it to be
mandated as "all."

Perhaps it should be.

Just now I looked through an issue of American Mathematical Monthly that
came to hand to get a sense of the likely alignment of Bourwiki with
what the mathematical community generally regards as the scope of its
subject.  Actually I do this periodically, and I don't see much change
between the issue I picked up just now and any of the other issues I've
looked at in the past with just this question in mind.

If the subject Bourwiki is proposing to serve is mathematics, then
perhaps it is time that the American Mathematical Monthly, along with
the Putnam Mathematical Competition, the International Mathematics
Olympiad, and the Journal of the AMS, abandon their pretense of being
about mathematics and come up with a suitable name for their subject.
Not only do categories, functors, natural transformations, adjunctions,
and monads go unused in these 20th century icons of mathematics, they go
unacknowledged.  Clearly they have not gotten with the modern
mathematical program and fall somewhere between a throwback to a golden
age and a backwater of mathematics.  When they die off like the
dinosaurs they are, real mathematics will be able to advance unfettered
into the 21st century and beyond.

Judging from the talks at BLAST in Denver last month (B = Boolean
algebras, L = lattices, A = (universal) algebra, S = set theory, T =
topology), at least the algebraic community is moving very slightly in
this direction.  Things will hopefully improve yet further when
algebraic geometry gets over its snit with equational model theory.

Meanwhile if you need a witness for seven degrees of separation, look no
further than AMM and CT.

(I confess to being an unreconstructed graph theorist and algebraist
myself.  I may have to preemptively volunteer myself for re-education
before it becomes involuntary.)

Vaughan Pratt

Re: Bourbaki and Categories

[Andrej Bauer]

Dear friends,

the usual kind of wiki is not suitable for collaborative science, but
recently there has been news of a special wiki for scientists which has
good support for references, keeps track of who said what, and has a
rating system. You can read more about it in Nature Genetics here

http://www.nature.com/ng/journal/v40/n9/full/ng.f.217.html

and see it working here: http://www.wikigenes.org/

They even have movies for those who are too lazy to click:

http://www.wikigenes.org/app/info/movie.html

It looks however that they are not offering the software that runs the
whole thing.

The next Bourbaki, if there is going to be one, should not only advance
one particular kind of knowledge, but also show everyone that linearly
written text is not the only option.

My opinion is that we have not yet found the right way to do
"hive-science", but when we do, it will be a revolution. (A good start
would be to get out of the hold that the evil publishers have on us.)

Best regards,

Andrej

Re: Bourbaki and Categories

[George Janelidze]

Dear Andree,

Could you please explain this better?:

The only Bourbaki member I new personally was Sammy Eilenberg. As many of
us, I knew him very well and I would say that he was more skeptical about
the Bourbaki Tractate then one can conclude from Andre's message. Having in
mind not just this but the content of Bourbaki's "Homological algebra" and
what we see today from the followers of that Bourbaki group, I protest
against Andre's "two options" and I insist that Bourbaki group simply did
not see the importance of category theory (in spite of being brilliant
mathematicians, as I said in my previous message). I hope Andre will forgive
me and even agree with me.

However, there were three great category-theorists in that group (plus there
is this mysterious story about Chevalley's book of category theory lost in
the train), and "did not see" cannot be said about them of course. On the
other hand I have never heard of any joint work of Charles Ehresmann with
any of the two others, Eilenberg and Grothendieck (and nothing jointly from
them). I think apart from the time issues you describe, the relationship
between Bourbaki Tractate and category theory should have been determined by
their separate or joint influence and therefore also by their communication
with each other (if any).

Is this true, and could you please give details?

Respectfully, and with best regards-

George

Re: Bourbaki and Categories

[Dusko Pavlovic]

i think that we should try to heed andre joyal's call for action. he calls
for a new collaborative effort a la bourbaki, this time based on
categories from the outset.

it is true that very ambitious efforts usually fail, and this would be an
extremely ambitious one. moreover, taking action sounds like something
people used to do in 20th century, and not in these times of fox news and
smooth crowd control.

but there are two points that make me think that andre's call is
different:

1) he is pointing to the reasons for action, that are slowly but surely
catching up with every scientist, no matter how much we try to ignore
them.

2) he is suggesting a medium (web, internet) that may make a difference
between... well between being able to make a difference and not being able
to make a difference.

ad (2), i would like to add that the web tools facilitate in a substantial
way not only dissemination, but also collaboration. there are methods to
support more efficient knowledge aggregation from a broader base than ever
before. developing a suitable collaboration process may be hard (at least
as hard as developing a suitable voting procedure), but it may be worth
while. eg, the wikipedia process can be criticized from many angles; but
wikipedia has the amazing property that it is an *evolutionary* knowledge
repository, which can easily correct any observed shortcomings, and
recover from any misinterpretations, almost like science itself.

at the moment, the wikipedia process is probably not optimal for
presenting subtle or many faceted concepts, and the discussions of
everyone with everyone else are not the most productive way. that is
perhaps why most of us (with some very honorable exceptions!) have been
staying away from it. but an improved process, combining the integrity,
and perhaps the structure of the categories@mta community with the
available wiki-methods may bring categorical methods into a dynamic
environment, perhaps more natural for them than books and papers.

just my 2c,
-- dusko

PS like an unwanted pop song, the name Nicolas Bourwiki just emerged in my
head! can someone please propose a worse one, or i am stuck. oh, i
already have a worse one...

On Sep 13, 2008, at 10:17 AM, Andre Joyal wrote:

> Bourbaki is dead but I hope that the humanistic philosophy behind the
> enterprise is not. Unfortunately, we presently live in an era of
> growing irrationalism.
> Science still needs to be defended against religion.
> Civilisation maybe at a turning point with the problem of climate
> change.
> Millions of people need and want to learn science and mathematics.
>
> Should we not try to give Bourbaki a second life?
> It will have to be different this time.
> Possibly with a new name.
> Obviously, internet is the medium of choice.
> What do you think?
>
> Andre

Re: Bourbaki and Categories

[David Spivak]

I agree with Andre.  Encapsulating a group of mathematicians inside a  
single named entity fosters a kind of collaborative spirit in which  
good ideas are not kept for personal use later but are shared amongst  
the community.  When ideas are shared in real time, good mathematics  
can be produced faster.  Anyone who wants to join the collective can  
do so, and the collective produces highly useful material.  Of course  
such an enterprise is orthogonal to name-recognition, and maybe to  
getting tenure!  But there is certainly something good about it, as  
there is about wikipedia and the open source movement.

I also agree that the internet could be used in a better way to  
transfer knowledge of mathematics.  Math papers are written linearly,  
in the bottom-up (Euclid/Bourbaki) style, to some extent.  Whereas  
words on paper are in this sense one-dimensional, computers offer  
many more dimensions for knowledge transfer.

Even more interesting to me would be a kind of zoom-feature on  
proofs.  Proofs are in the eye of the beholder: for example it has  
been debated as to whether Perelman's 70 pages was a full proof of  
geometrization.  Given a proof with a statement which one does not  
understand, a mathematician may find himself reproving something that  
was obvious to (or wrongly assumed to be obvious by) another  
mathematician.  The community could benefit if a mathematician who  
proves such a statement then uploaded the proof, even in rough form,  
to some kind of math wiki.  If it were well-organized, this math wiki  
could revolutionize how mathematics is done.  In fact, choosing the  
"right way" to organize such a site may itself be a problem which  
could produce interesting mathematics.

Whatever the case may be, I am all for the idea of a new Bourbaki- 
style enterprise in some form or another.  I think it may first  
require interested parties to get together at some physical location.

David


On Sep 13, 2008, at 10:17 AM, Andre Joyal wrote:

> Dear Colin, Zoran, Robert, Eduardo and All,
>
> I find the present discussion on Bourbaki and category theory very  
> important.
> I recall asking the question to Samuel Eilenberg 25 years ago and  
> more recently to Pierre Cartier.
> If my recollection is right, Bourbaki had essentially two options:  
> rewrite the whole treaty using categories,
> or just introduce them in the book on homological algebra,
> The second option won, essentially because of the enormity of the  
> task of rewriting everything.
> Other factors may have contributed on a smaller scale, like some  
> unresolved foundational questions.
> In any cases, it was the beginning of end for Bourbaki.
>
> Bourbaki was a great humanistic and scientific enterprise.
> Advanced mathematics was made available to a large number
> of students, possibly over the head of their bad teachers.
> It defended the unity and rationality of science in an age
> of growing irrationalism (it was conceived in the mid thirties).
>
> I have personally learned a lot of mathematics by reading Bourbaki.
> Everything was proved, and the proofs were logically very clear.
>  It was a like a continuation of Euclid Elements two thousand years  
> later!
> But after a while, I stopped reading it.
> I had realised that something important was missing: the motivation.
> The historical notes were very sketchy and not integrated to the text.
> I remember my feeling of frustration in reading the books of  
> functional analysis,
> because the applications to partial differential equations were not  
> described.
> Everything was presented in a deductive order, from top to down.
> We all know that learning is very much an inductive process, from
> the particular to the general. This is true also of mathematical  
> research.
>
> Bourbaki is dead but I hope that the humanistic philosophy behind  
> the enterprise is not.
> Unfortunately, we presently live in an era of growing irrationalism.
> Science still needs to be defended against religion.
> Civilisation maybe at a turning point with the problem of climate  
> change.
> Millions of people need and want to learn science and mathematics.
>
> Should we not try to give Bourbaki a second life?
> It will have to be different this time.
> Possibly with a new name.
> Obviously, internet is the medium of choice.
> What do you think?
>
> Andre
>
>

Re: Bourbaki and Categories

[Michael Barr]

I don't know about this.  I took several courses in the late 1950s that
seem to have been influenced by the structuralist ideas (certainly
categories weren't mentioned; I never heard the word until Dave Harrison
arrived in 1959) and each of them started by defining an appropriate
notion of "admissible map".  I do not recall any special point being made
of isomorphism and I think in general it was used for what we now call a
bimorphism (1-1 and onto) even in cases, such as topological groups, when
they were not isomorphisms.

To be sure Bourbaki was not mentioned either, but this structuralist
influence seemed strong.

Michael

On Mon, 15 Sep 2008, Andre.Rodin@ens.fr wrote:

>
> zoran skoda wrote:
>
>
>> The remark that as a proponent of "structures"
>> Bourbaki had to include categories is anyway a bit lacking an argument.
>
>
>
> I think that as a 'proponent of "structures"' Bourbaki had NOT include
> categories - and not only because of the size problem. A more fundamental
> reason seems me to be this. Structures are things determined up to isomorphism;
> in the structuralist mathematics the notion of isomorphism is basic and the
> notion of general morphism is derived (as in Bourbaki).  In CT this is the
> other way round: the notion of general morphism is basic while isos are defined
> through a specific property (of reversibility).
> This is why the inclusion of CT would require a revision of fundamentals of
> Bourbaki's structuralist thinking. Although CT for obvious historical reasons
> is closely related to structuralist mathematics it is not, in my understanding,
> a part of structuralist mathematics - at least not if one takes CT *seriously*,
> i.e. as foundations.
>
> best,
> andrei
>
>
>

Re: Bourbaki and Categories

[Andree Ehresmann]

  Dear all,

I would add some information on Bourbaki/categories/France.
Charles Ehresmann has been an active member of Bourbaki from 1936 up  
to the end of the war, when he began to no more regularly participate  
and wanted to resign (it was not accepted but replaced by an age limit  
for active participation).
What Andre says:

    >Bourbaki had essentially two options: rewrite the whole treaty using
    > categories, or just introduce them in the book on homological algebra,
    >The second option won, essentially because of the enormity of the task
    > of rewriting everything.

is more easily understood if we take into account that communication  
between France and the USA were entirely broken during the war, so  
that mathematical ideas could not circulate and categories were only  
heard of after the war, at a time where the more general parts of the  
treatise were written or at least prepared (the successive versions  
process was very slow).
     Charles said to me that he did not recall to have read Eilenberg  
& Mac Lane's paper before the fifties, or at least not seen its  
interest. Naturally he had sooner made a large use of groupoids in is  
foundation of differential geometry, and he had even defined the  
general "composition of jets" and given its properties, but without  
linking it to the notion of a category. He exposed it in a course in  
Rio de Janeiro in the early fifties, and one of his students  
(Constantino de Barros who later came to Paris to prepare a thesis  
with him) suggested that there was a connection with categories.  
Charles' first large use of categories is in his seminal paper  
"Gattungen von lokalen Strukturen" (1957, reprinted in "Charles  
Ehresmann: Oeuvres completes et commentees" Part I).
    It is around this date that the word "category" began to circulate  
in France.  In 1957, Choquet  (with whom I prepared my thesis)  
suggested that I learnt more on the notion of category which he did  
not know but seemed to have many applications (it was the reason for  
which I first went to see Charles!). It should be noted that Choquet  
was less conservative than many French mathematicians. In 1959, he  
defended the development of probabilities by inviting Loomis to give a  
course (I remember Henri Cartan saying then to Paul-Andre Meyer that  
he should not study this domain for it would be bad for his career!).  
And later on, he defended Logic which was very badly considered.
    A final remark: the "disdain" for categories (not to be confused  
with 'ignorance') came only later on, since Charles was given the  
"Prix Petit d'Ormoy" by the French Academy in 1965, essentially for  
his recent work on categories...

Andree C. Ehresmann

Re: Bourbaki and Categories

[Andre.Rodin]

zoran skoda wrote:


>The remark that as a proponent of "structures"
>Bourbaki had to include categories is anyway a bit lacking an argument.



I think that as a 'proponent of "structures"' Bourbaki had NOT include
categories - and not only because of the size problem. A more fundamental
reason seems me to be this. Structures are things determined up to isomorphism;
in the structuralist mathematics the notion of isomorphism is basic and the
notion of general morphism is derived (as in Bourbaki).  In CT this is the
other way round: the notion of general morphism is basic while isos are defined
through a specific property (of reversibility).
This is why the inclusion of CT would require a revision of fundamentals of
Bourbaki's structuralist thinking. Although CT for obvious historical reasons
is closely related to structuralist mathematics it is not, in my understanding,
a part of structuralist mathematics - at least not if one takes CT *seriously*,
i.e. as foundations.

best,
andrei

Re: Bourbaki and Categories

[mjhealy]

Andre Joyal's message was inspiring.  I think a new Bourbaki type of
effort (this time with motivation and category theory) is called for.  I
would like a text on categorical algebra oriented toward those who have
studied algebra and know enough category theory to use adjunctions, monads
and the like.  The same goes for categorical logic and model theory.

I read Andre's closing remarks as a commentary on the emerging crisis in
overcoming misconceptions about and outright hostility toward science and
mathemtics.  I have a current project, using my own meager funds and time,
to advance science teaching using a book available through the National
Academies Press, "Science, Evolution, and Creationism".  From the title, I
think you can see my motivation.

In a similar (although less aprocryphal) vein, when I worked with computer
scientists and applied mathematicians in industry ( and also when I've
submitted papers to certain neural network journals) I encountered
misconceptions about and outright hostility toward category theory.  For
example, in the dynamic systems community there seems to be a widespread
myth that "category theory was tried and failed".  I have followed this up
to some extent and haven't found any basis for it.  I am often told that
the best way to counter skepticism is with a working application.

Having tried that, and tried again, I've come to the conclusion that Yes,
you need applications, but applications cannot by themselves counter a
refusal to give a theory credit for being consistent with the data.  You
need a good, clear presentation of the theory couched in a language
oriented toward the intended audience.  As with biology teaching that
shows clearly the importance of the theory of evolution, maybe mathematics
teaching that incorporates category theory needs to begin in 6th Grade (in
schools in the USA) if not sooner.  Maybe a new Bourbaki project could
have an extension into this level of instruction.

Best Regards,
Mike


Please excuse my deviating from mathematics

Re: Bourbaki and Categories

[R Brown]

Dear All,

The importance of Bourbaki should be stessed, as it was started when, so we 
are told, texts were very bad. There are many beautiful things in the books: 
I developed part of an undergraduate course from the account of the 
classification of closed subgroups of R^n. This relates to old questions on 
orbits of the planets, and also gives some nice exercises and even exam 
questions of a calculation type. It is good to present students with a 
classification theorem.

The difficulties for Bourbaki seem to arise from the presentation (a) as a 
final and definitive view in toto, and (b) without enough context, as Andre 
points out.

On (a), there is the old childish joke: what happens if you put worms in a 
straight line from Marble Arch to Picadilly Circus? One of them would be 
bound to wriggle and spoil it all! So some mathematical worms have not only 
wriggled but grown large and marched off in a different direction.

On (b), there is the old debating society tag:
text without context is merely pretext.
See more questions in Tim and my article on `Mathematics in Context'.

What is wrong is to present, or take, the whole account as totally 
authoritative, and  will last indefinitely.

What Bourbaki also shows is the value for at least the writers of taking a 
viewpoint and following it through as far as it will go: if it seems in the 
end to go too far, or to be inadequate, then that is valuable information 
for them and others. See my Dirac quote in `Out of Line'.

Ronnie








----- Original Message ----- 
From: "Andre Joyal" <joyal.andre@uqam.ca>
To: <categories@mta.ca>
Sent: Saturday, September 13, 2008 6:17 PM
Subject: categories: Re: Bourbaki and Categories


Dear Colin, Zoran, Robert, Eduardo and All,

I find the present discussion on Bourbaki and category theory very 
important.
I recall asking the question to Samuel Eilenberg 25 years ago and more 
recently to Pierre Cartier.
If my recollection is right, Bourbaki had essentially two options: rewrite 
the whole treaty using categories,
or just introduce them in the book on homological algebra,
The second option won, essentially because of the enormity of the task of 
rewriting everything.
Other factors may have contributed on a smaller scale, like some unresolved 
foundational questions.
In any cases, it was the beginning of end for Bourbaki.

Bourbaki was a great humanistic and scientific enterprise.
Advanced mathematics was made available to a large number
of students, possibly over the head of their bad teachers.
It defended the unity and rationality of science in an age
of growing irrationalism (it was conceived in the mid thirties).

I have personally learned a lot of mathematics by reading Bourbaki.
Everything was proved, and the proofs were logically very clear.
 It was a like a continuation of Euclid Elements two thousand years later!
But after a while, I stopped reading it.
I had realised that something important was missing: the motivation.
The historical notes were very sketchy and not integrated to the text.
I remember my feeling of frustration in reading the books of functional 
analysis,
because the applications to partial differential equations were not 
described.
Everything was presented in a deductive order, from top to down.
We all know that learning is very much an inductive process, from
the particular to the general. This is true also of mathematical research.

Bourbaki is dead but I hope that the humanistic philosophy behind the 
enterprise is not.
Unfortunately, we presently live in an era of growing irrationalism.
Science still needs to be defended against religion.
Civilisation maybe at a turning point with the problem of climate change.
Millions of people need and want to learn science and mathematics.

Should we not try to give Bourbaki a second life?
It will have to be different this time.
Possibly with a new name.
Obviously, internet is the medium of choice.
What do you think?

Andre




-------- Message d'origine--------
De: cat-dist@mta.ca de la part de Colin McLarty
Date: ven. 12/09/2008 14:46
À: categories@mta.ca
Objet : categories: Re:  Bourbaki and Categories

From: zoran skoda <zskoda@gmail.com>
Date: Friday, September 12, 2008 2:06 pm

wrote, among other things

> main points of departure. The remark that as a proponent of
> "structures" Bourbaki
> had to include categories is anyway a bit lacking an argument.
> First of all, because
> of the size problems one can not take big categories on equal
> footing with, say groups,
> and considering only small categories would be strange and lacking
> most interesting examples.

The claim is not that Bourbaki should have studied categories as
structures.  It is that Bourbaki was doomed to fail in trying to use
their structure theory.  Leo Corry shows in his book "Modern Algebra and
the Rise of Mathematical Structures" (Birkhäuser 1996) that they did fail.

And they should have seen this coming, because their theory had been

"superseded by that of category and functor, which includes it under a
more general
and convenient form" (Dieudonné "The Work of Nicholas Bourbaki" 1970).

best, Colin

Re: Bourbaki and Categories

[Andre Joyal]

Dear Colin, Zoran, Robert, Eduardo and All,

I find the present discussion on Bourbaki and category theory very important.  
I recall asking the question to Samuel Eilenberg 25 years ago and more recently to Pierre Cartier.  
If my recollection is right, Bourbaki had essentially two options: rewrite the whole treaty using categories, 
or just introduce them in the book on homological algebra, 
The second option won, essentially because of the enormity of the task of rewriting everything. 
Other factors may have contributed on a smaller scale, like some unresolved foundational questions. 
In any cases, it was the beginning of end for Bourbaki.

Bourbaki was a great humanistic and scientific enterprise.
Advanced mathematics was made available to a large number
of students, possibly over the head of their bad teachers. 
It defended the unity and rationality of science in an age
of growing irrationalism (it was conceived in the mid thirties).

I have personally learned a lot of mathematics by reading Bourbaki.  
Everything was proved, and the proofs were logically very clear.
 It was a like a continuation of Euclid Elements two thousand years later!
But after a while, I stopped reading it.
I had realised that something important was missing: the motivation. 
The historical notes were very sketchy and not integrated to the text.
I remember my feeling of frustration in reading the books of functional analysis,
because the applications to partial differential equations were not described.
Everything was presented in a deductive order, from top to down.
We all know that learning is very much an inductive process, from
the particular to the general. This is true also of mathematical research. 

Bourbaki is dead but I hope that the humanistic philosophy behind the enterprise is not.  
Unfortunately, we presently live in an era of growing irrationalism.
Science still needs to be defended against religion.
Civilisation maybe at a turning point with the problem of climate change. 
Millions of people need and want to learn science and mathematics. 

Should we not try to give Bourbaki a second life? 
It will have to be different this time.
Possibly with a new name.
Obviously, internet is the medium of choice.
What do you think?

Andre




-------- Message d'origine--------
De: cat-dist@mta.ca de la part de Colin McLarty
Date: ven. 12/09/2008 14:46
À: categories@mta.ca
Objet : categories: Re:  Bourbaki and Categories
 
From: zoran skoda <zskoda@gmail.com>
Date: Friday, September 12, 2008 2:06 pm

wrote, among other things

> main points of departure. The remark that as a proponent of 
> "structures" Bourbaki
> had to include categories is anyway a bit lacking an argument. 
> First of all, because
> of the size problems one can not take big categories on equal 
> footing with, say groups,
> and considering only small categories would be strange and lacking 
> most interesting examples.

The claim is not that Bourbaki should have studied categories as
structures.  It is that Bourbaki was doomed to fail in trying to use
their structure theory.  Leo Corry shows in his book "Modern Algebra and
the Rise of Mathematical Structures" (Birkhäuser 1996) that they did fail.  

And they should have seen this coming, because their theory had been 

"superseded by that of category and functor, which includes it under a
more general
and convenient form" (Dieudonné "The Work of Nicholas Bourbaki" 1970).

best, Colin

Re: Bourbaki and Categories

[George Janelidze]

Dear Colleagues,

I think the first things to say about "Bourbaki and Categories" are:

(a) It is very obvious that the invention of category theory was by far the
greatest discovery of 20th century mathematics.

(b) Bourbaki Tractate is another great event, of a very different kind of
course, which will be a treasure for the Historians of next centuries. It
shows how the members of a very leading group of a leading mathematical
country were thinking in the middle of the same century (well, up to their
internal disagreements; after all, Eilenberg and Grothendieck were also
there at some point...).

(c) Accordingly, Bourbaki Tractate is the best evidence showing how hard it
was to understand (even and especially for such brilliant mathematicians!)
that there is something even better that Cantor paradise.

(d) Defining structures, Bourbaki makes very clear that morphisms are
important (and some form of universal properties are important). But
morphisms are NOT defined in general: it is simply a class of maps between
structures of a given type closed under composition and having isomorphisms
(which ARE defined) as its invertible members. And... every interested
student will ask: if so, why not defining a category?

Let me also add what is less important but still comes to my mind:

(e) Bourbaki approach to structures has a hidden very primitive form of what
was later discovered by topos theorists: in order to define a structure they
need a 'scales of sets', which is build using finite products and power sets
(no unions and no colimits of any kind!).

(f) According to Walter Tholen's message, Karl Heinrich Hofmann says: "...it
is truly surprising that the theory of categories (S. Eilenberg and S. Mac
Lane, 1946) was almost demonstrably
ignored as the mother of all structure theories. This was hardly sustainable
in commutative algebra anymore...". Very true (except 1946), but it is
much-much-much worse in homological algebra, where the absence of categories
and functors (having a section called "Functoriality" though) in Bourbaki's
presentation is most amazing.

(g) A few days ago Tom Leinster has explained to us that "disinformation is
*deliberate* false information, false information *intended* to mislead".
Fine, but sometimes false information is created by ignorance so badly, that
it sounds right to call it disinformation (Don't you agree, Tom?). And...
look at http://en.wikipedia.org/wiki/Bourbaki : There is a section called
"Criticism of the Bourbaki perspective", which, among other things, says:
"The following is a list of some of the criticisms commonly made of the
Bourbaki approach:^[13]..." (where [13] is a book of Pierre Cartier; I have
not seen that book, and so I am not making any conclusions about it). The
list has seven items with no category theory in it!

George Janelidze

Re: Bourbaki and Categories

[Colin McLarty]

From: Robert Seely <rags@math.mcgill.ca>
Date: Friday, September 12, 2008 8:17 pm

noted

> For those who like compliments: the triples website has had (for a
> while now) a link to a lecture by Voevodsky given at the American
> Academy of Arts and Sciences in October 2002, in which he describes
> "categories [as] one of the most important ideas of 20th century
> mathematics".  The video of the talk may be found at
>
>   http://claymath.msri.org/voevodsky2002.mov
>
> (the compliment isn't the only reason for watching!).


It is a terrific lecture.  The line "I think that at the heart of 20th
century mathematics lies one particular notion and that is the notion
of a category" occurs at minute 16.

best, Colin

Re: Bourbaki and Categories

[Robert Seely]

For those who like compliments: the triples website has had (for a
while now) a link to a lecture by Voevodsky given at the American
Academy of Arts and Sciences in October 2002, in which he describes
"categories [as] one of the most important ideas of 20th century
mathematics".  The video of the talk may be found at

   http://claymath.msri.org/voevodsky2002.mov

(the compliment isn't the only reason for watching!).

And there are a few other categorical links on our site at

  http://www.math.mcgill.ca/triples/

Suggestions are always welcome.

-= rags =-


-- 
<rags@math.mcgill.ca>
<www.math.mcgill.ca/rags>

Re: Bourbaki and Categories

[Colin McLarty]

From: zoran skoda <zskoda@gmail.com>
Date: Friday, September 12, 2008 2:06 pm

wrote, among other things

> main points of departure. The remark that as a proponent of 
> "structures" Bourbaki
> had to include categories is anyway a bit lacking an argument. 
> First of all, because
> of the size problems one can not take big categories on equal 
> footing with, say groups,
> and considering only small categories would be strange and lacking 
> most interesting examples.

The claim is not that Bourbaki should have studied categories as
structures.  It is that Bourbaki was doomed to fail in trying to use
their structure theory.  Leo Corry shows in his book "Modern Algebra and
the Rise of Mathematical Structures" (Birkhäuser 1996) that they did fail.  

And they should have seen this coming, because their theory had been 

"superseded by that of category and functor, which includes it under a
more general
and convenient form" (Dieudonné "The Work of Nicholas Bourbaki" 1970).

best, Colin

Re: Bourbaki and Categories

[zoran skoda]

>
> sustainable in commutative algebra anymore, and the discord between
> Grothendieck and Bourbaki may well have been rooted in this rejection.


I can not recall where, but I read more than once more detailed descriptions
on
what Bourbaki did not accept from Grothendieck. The conservativeness
of Bourbaki who did not accept the usage of category theory (not only
"neglect")
and non-acceptance of a very general approach of Grothendieck to the
notion of "manifold" he envisioned for the future Bourbaki works
were some of the
main points of departure. The remark that as a proponent of "structures"
Bourbaki
had to include categories is anyway a bit lacking an argument. First of all,
because
of the size problems one can not take big categories on equal footing with,
say groups,
and considering only small categories would be strange and lacking most
interesting
examples. On the other hand, Grothendieck judged the lack stemming in
conservativeness
rather than in consistency of the structure-oriented style. Indeed,
according to Dieudonne,
Bourbaki felt comfortable only in including to the books already
(meta)stable, "dead" mathematics
and not the structures in the unstable "living" phase of development.
This was the intended scope and self-conscious (according to Dieudonne)
limitation of the work.
One can accept this and still cry for an exception for so economic tool
as the category theory (if taken in conservative and very basic sense),
especially
in the vision of the wish for generality, Bourbaki followed otherwise.

Zoran Skoda

Bourbaki and Categories

[Walter Tholen]

Here is some up-lifting press about categories that I saw in an article
by Karl  Heinrich Hofmann entitled "Bourbaki in T"ubingen und in den
USA, Erinnerungen an die franz"osische Revolution in der Mathematik",
which may translate as "Bourbaki in Tubingen and in the USA,
reminiscenses of the French revolution in mathematics", and which
appeared in the "Mitteilungen der DMV" (the German equivalent of the AMS
Notices, which is distributed to all members), vol 16.2 (2008),
pp128-136. While the author has a lot of praise for Bourbaki's work, he
lists also a number of "defects of the Bourbaki concept", and the
following appears quite prominently in his article (my translation,
okayed by the author):

"Since Bourbaki is considered as the exponent of the theory of
mathematical structures, it is truly surprising that the theory of
categories (S. Eilenberg and S. Mac Lane, 1946) was almost demonstrably
ignored as the mother of all structure theories. This was hardly
sustainable in commutative algebra anymore, and the discord between
Grothendieck and Bourbaki may well have been rooted in this rejection.
This dismissive position is even more surprising since Eilenberg as one
of the few non-French people belonged to the early Bourbaki group, and
since the French founder of category theory, Charles Ehresmann, was at
times closely connected with Bourbaki. In my view this failure of
Bourbaki is grave."