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I am trying to understand modelchecking logics such as=20
linear temporal logic (LTL), computation tree logic (CTL),
and probabilistic extensions such as probabilistic computation
tree logic (PCTL). These logics seem to be useful in model
checking because of correspondences between recursive=20
definitions of formulas and recursive algorithms for
computing them. Syntactic inference and any semantics
other than the few (transition systems, DTMCs, MDPs, etc.) for=20
which the logics were developed seem not to be of interest,=20
however. Translation theorems such as D1.4.7, D1.4.12, and
D4.3.13 of Johnstone's "Elephant" are not applied, as far as=20
I have found.
Here are some initial thoughts based on my very limited=20
knowledge of categorical logic. The temporal logic operators=20
"next", "eventually", and "always" may, for example,=20
be formulated using a propositional signature. Let A be a set=20
whose elements are thought of as state labels. The set of=20
atomic propositions is {a_n  a in A, n in N}. The following=20
are Horn formulas:
Next^n[a] =3D a_n
Always^k[a] =3D a_0 \wedge ... \wedge a_k
T^k[a, b] =3D a_0 \wedge ... \wedge a_{k1} \wedge b_k
The following are coherent formulas:
Until^k[a, b] =3D T^0[a, b] \vee ... \vee T^k[a, b]
Eventually^k[a] =3D a_0 \vee ... \vee a_k
The following are geometric formulas:
Until[a, b] =3D \vee_i T^i[a, b]
Eventually[a] =3D \vee_i a_i
The following is infinitary:
Always[a] =3D \wedge_i a_i.
Alternatively, these notions can be formulated as a=20
firstorder theory with basic sorts "States" and "Paths". =20
It has function symbols
start : Paths > States
shift : Paths > Paths
and a set A of of relation symbols=20
a > States.
PCTL can be formulated as a \tautheory. It has basic sorts=20
States and Paths as above as well as the function symbols=20
start and shift. It also has function symbols=20
Prob_{ P(States)=20
(similarly for \leq, >, and \geq) for 0 \leq p\leq 1 and
Cost_{ P(S)=20
(similarly for \leq, >, and \geq) for 0 \leq c.
There are certainly other presentations of these theories. =20
Finally, here are a few questions.
Has anyone studied the modelchecking logics using the tools
of categorical logic?
What interactions exist between bisimulation/probabilistic=20
bisimulation and the logics?
What role could categorical logic have in developing model
checking algorithms?
How can the usual semantics of these theories (e.g., DTMC=20
semantics for PCTL) be viewed categorically (e.g., in the=20
topos Set by interpreting the sorts simply as sets of states
and paths)?
Any other thoughts, suggestions, or references would be=20
appreciated.
References:
Arnold. "Finite Transition Systems". PrenticeHall. 1994.
Johnstone. "Sketches of an Elephant: A Topos Theory Compendium". =20
Oxford University Press. 2002.
Rutten, Kwiatkowska, Norman, and Parker. "Mathematical Techniques
for Analyzing Concurrent and Probabilistic Systems". AMS. 2004.
Thanks,
Ralph Wojtowicz
Metron, Inc.
11911 Freedom Drive, Suite 800
Reston, VA USA
wojtowicz@metsci.com
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To: categories@mta.ca
Subject: categories: Domains IX [Final Call for Registration]
Date: Tue, 2 Sep 2008 18:12:05 +0100
From: Bernhard Reus
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Registration for the International Workshop

DOMAINS IX

on domain theory, denotational semantics and applications,
taking place from September 22nd  24th at the University of Sussex
is about to close now.
Registration forms must be received by Friday, September 5th, 6am BST.
More information on the workshop and how to register can be found at:
http://www.informatics.sussex.ac.uk/events/domains9/index.htm
Regards,
Bernhard
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Date: Tue, 2 Sep 2008 15:00:39 0700
From: John Baez
To: categories
Subject: categories: Re: KT Chen's smooth CCC
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Bill Lawvere wrote:
>By urging the study of the good geometrical ideas and constructions of
>Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spanier,
>Steenrod, I am of course not advocating the preferential resurrection of
>the particular categories they tentatively devised to contain the
>constructions.
I chose Chen's framework when Urs Schreiber and I were doing some work
in mathematical physics and we needed a "convenient category" of smooth
spaces. I decided to choose one that was easy to explain to people
brainwashed by the "default paradigm", in which spaces are sets equipped
with extra structure. Later I realized I needed to write a paper
establishing some properties of Chen's framework. By doing that I guess
I'm guilty of reinforcing the default paradigm, and for that I apologize.
If I understand correctly, one can actually separate the objections
to continuing to develop Chen's theory of "differentiable spaces"
into two layers.
Let me remind everyone of Chen's 1977 definition. He didn't state
it this way, but it's equivalent:
There's a category S whose objects are convex subsets C of R^n
(n = 0,1,2,...) and whose maps are smooth maps between these.
This category admits a Grothendieck pretopology where a cover
is an open cover in the usual sense.
A differentiable spaces is then a sheaf X on S. We think of
X as a smooth space, and X(C) as the set of smooth maps from C to X.
But the way Chen sets it up, differentiable spaces are not all
the sheaves on S: just the "concrete" ones.
These are defined using the terminal object 1 in S. Any convex set
C has an underlying set of points hom(1,C). Any sheaf X on S has an
underlying set of points X(1). Thanks to these, any element of X(C)
has an underlying function from hom(1,C) to X(1). We say X is "concrete"
if for all C, the map sending elements of X(C) to their underlying
functions is 11.
The supposed advantage of concrete sheaves is that the underlying
set functor X > X(1) is faithful on these. So, we can think of
them as sets with extra structure.
But this advantage is largely illusory. The concreteness condition
is not very important in practice, and the concrete sheaves form not
a topos, but only a quasitopos.
That's one layer of objections. Of course, *these* objections
can be answered by working with the topos of *all* sheaves on S.
This topos contains some useful nonconcrete objects: for example,
an object F such that F^X is the 1forms on X.
But now comes a second layer of objections. This topos of sheaves
still lacks other key features of synthetic differential geometry.
Most importantly, it lacks the "infinitesimal arrow" object D such
that X^D is the tangent bundle of X.
The problem is that all the objects of S are ordinary "noninfinitesimal"
spaces. There should only be one smooth map from any such space to D.
So as a sheaf on S, D would be indistinguishable from the 1point space.
So I guess the real problem is that the site S is concrete: that is,
the functor assigning to any convex set C its set of points hom(1,C)
is faithful. I could be jumping to conclusions, but it seems to me
that that sheaves on a concrete site can never serve as a framework
for differential geometry with infinitesimals.
Best,
jb
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Date: Tue, 02 Sep 2008 14:07:31 0400
From: Claudio Hermida
To: categories@mta.ca
Subject: categories: Re: logics for model checking
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Wojtowicz, Ralph wrote:
> Finally, here are a few questions.
>
> Has anyone studied the modelchecking logics using the tools
> of categorical logic?
>
>
Here's a little note I wrote about relational modalities from a
categorical logic viewpoint; it includes a few further references which
may be relevant to your question
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.18.3240
Claudio Hermida
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Date: Wed, 03 Sep 2008 22:59:53 +0200
From: Luigi Santocanale
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Hi,
> Any other thoughts, suggestions, or references would be
> appreciated.
In
L. S. Completions of \mualgebras. APAL, 154(1):2750, May 2008.
I studied the problem of the completeness of the modal mucalculus from=20
an finitary algebraic point of view. In that paper some categorical=20
ideas, mainly from
W. Tholen, Procategories and multiadjoint functors, Canad. J. Math. 36=20
(1) (1984) 144=96155.
play the relevant role. The challenge is to prove that in free modal=20
\mualgebras, the relation
\mu.f =3D \bigvee_{n>=3D0} f^n(\bot)
holds  where \mu.f, the least fixpoint of f, is axiomatized by=20
equational implications and free modal \mualgebras are not known to be=20
complete.
Best,
Luigi
=20
Luigi Santocanale
LIF/CMI Marseille T=E9l: 04 91 11 35 74
http://www.cmi.univmrs.fr/~lsantoca/ Fax: 04 91 11 36 02 =09
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for categorieslist@mta.ca; Wed, 03 Sep 2008 19:49:18 0300
Date: Wed, 3 Sep 2008 17:57:21 0400 (EDT)
From: Michael Barr
To: Categories list
Subject: categories: Octoberfest
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We have received a bit of money from Concordia Univeristy towards the
Octoberfest and have decided that we would like to help out graduate students
who wish to attend. So any grad student who is planning to attend and would
like support should write to Robert Seely (rags@math.mcgill.ca) by Sept 15 at
the latest and make a request. We may or may not be able to help with travel
expenses, but we should be able to pay for local hotel bills, although
everything depends on how many applicants. Participants will have to
submit original bills to be reimbursed. Sorry, but it will be impossible
to provide money beforehand.
As usual, people wishing to talk should let us know ASAP. Since we are trying
to arrange a lunch on Saturday, we would also appreciate knowing who is
planning to come. Again ASAP.
Michael
From rrosebru@mta.ca Thu Sep 4 09:05:49 2008 0300
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for categorieslist@mta.ca; Thu, 04 Sep 2008 09:02:51 0300
Date: Thu, 04 Sep 2008 08:45:02 +0100
From: Maria Manuel Clementino
To: categories@mta.ca
Subject: categories: Research positions
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5YEAR RESEARCH POSITIONS
The Centre for Mathematics of the University of Coimbra (CMUC) invites
applications for five 5year research positions in all areas of Mathemati=
cs.
Applicants should have a PhD in Mathematics and at least three years of
postdoctoral experience, with an internationallevel publication record=20
and demonstrated ability to perform independent research. Preference=20
will be given to candidates working in one of the areas in which the=20
members of CMUC are currently active. A very good command of English is=20
required.
The appointments will be made for three years, renewable for two more.
Successful applicants are expected to develop research within the areas=20
of interest of CMUC. They should also contribute to graduate studies,=20
supervising students from the Department of Mathematics.
The basic salary is equivalent to the index 195 of the Portuguese
research career (in the order of 42 500 euros a year), in accordance=20
with the Ci=EAncia 2008 guidelines
http://alfa.fct.mctes.pt/apoios/contratacaodoutorados/edital2008
Benefits include private health insurance and social security.
Applicants should send
=95 cover sheet (available from www.mat.uc.pt/=18cmuc/coversheet.pdf),=20
including names and contact information of three persons who can provide=20
a letter of recommendation
=95 curriculum vitae (publication list included)
=95 a 5year research proposal (2 pages max), emphasizing their own
research goals as pdf files to cmuc@mat.uc.pt. The subject of the email=20
message must include the job reference C2008 FCTUC CMUC v15v19.
The deadline for applications is *September 10, 2008.*

More information can be found at www.mat.uc.pt
From rrosebru@mta.ca Fri Sep 5 07:41:17 2008 0300
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for categorieslist@mta.ca; Fri, 05 Sep 2008 07:35:33 0300
Date: Thu, 4 Sep 2008 16:22:34 0700
From: "Meredith Gregory"
To: categories@mta.ca
Subject: categories: language for infinitary compositions?
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Categorists,
Is there a commonly accepted language for infinitary compositions? Here's
the sort of thing i'm thinking about. There's a purely syntactic
correspondence between a 'braced' notation and an infix notation for
composition. Suppose we have a categoricallyfriendly notion of composition,
say c. (Meaning the entities c composes can be viewed as morphisms in a
category and c the categorical composition.) Then we can just as easily
write
 f c g  c is merely making notationally explicit the interpretation of
'o' in f o g  we're coloring the 'o', as it were
or
 {c f, g c}  we've moved from infix to (not quite) prefix notation
for the composition.
The braced notation, however, is suggestive of a very powerful notational
mechanism, comprehension notation. We could easily imagine a language
allowing expressions of the form
{c pattern  predicate c}
which would denote
pattern{subst_1} c pattern{subst_2} c ...
where subst_i is a substition for 'variables' in the pattern of entities
satisfying the predicate. This would allow reasoning over infinitary
compositions by providing an intensional view of their interior structure.
Surely, such a widget has already been invented. Can someone give me a
reference?
Best wishes,
greg

L.G. Meredith
Managing Partner
Biosimilarity LLC
806 55th St NE
Seattle, WA 98105
+1 206.650.3740
http://biosimilarity.blogspot.com
From rrosebru@mta.ca Sun Sep 7 14:13:33 2008 0300
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for categorieslist@mta.ca; Sun, 07 Sep 2008 14:06:28 0300
Date: Fri, 05 Sep 2008 11:11:51 +0100
To: categories@mta.ca
Subject: categories: New Association Computability in Europe formed
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From: A.Beckmann@swansea.ac.uk (Arnold Beckmann)
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After four very successful conferences in Amsterdam in 2005, Swansea in
2006, Siena in 2007 and Athens in 2008, our community has officially
formed the association
Computability in Europe
at the Annual General Meeting at this year's Computability in Europe
conference in Athens. The object of the Association is to promote the
development, particularly in Europe, of computabilityrelated science,
ranging over mathematics, computer science, and applications in various
natural and engineering sciences such as physics and biology. This also
includes the promotion of the study of philosophy and history of computing
as it relates to questions of computability. A draft constitution of the
Association can be found at
http://www.amsta.leeds.ac.uk/~pmt6sbc/CiE.const.draft.pdf
We invite every researcher interested in the object of the Association to
become a member. The initial membership fee is set at zero, and lasts
until 30 June 2010.
To apply for membership of the Association, please complete and submit the
form at
http://www.cs.swan.ac.uk/acie/
Any enquiries concerning association CiE membership should be sent to the
Membership Secretary, Arnold Beckmann, at a.beckmann@swansea.ac.uk.
If you are not interested in becoming a member of this Association, we
apologise for any inconvenience caused.
With best regards,
Association Computability in Europe
***********************************************************************
Dr Arnold Beckmann  Swansea University
 Computer Science

 a.beckmann@swansea.ac.uk
 http://www.cs.swan.ac.uk/~csarnold/
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for categorieslist@mta.ca; Sun, 07 Sep 2008 14:08:04 0300
Date: Fri, 5 Sep 2008 11:49:43 0400 (EDT)
From: Michael Barr
To: Categories list
Subject: categories: Amusing fact
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There is nothing deep in the following, but it is amusing and slightly
surprising. In the wellknown diagram below (the dots indicate spaces,
which email doesn't handle well):
........0........0.......0..........
.................................
.................................
.................................
........v........v.......v..........
0 > A' > A > A'' > 0
.................................
.................................
.................................
........v........v.... ..v..........
........ ... f .. .. g .. ..........
0 > B' > B > B'' > 0
.................................
.................................
.................................
........v........v.......v..........
0 > C' > C > C'' > 0
.................................
.................................
.................................
........v........v.......v..........
........0........0.......0..........
it is widely known that if the three columns, middle row and one of the
other two rows is exact, so is the remaining row. What if the upper and
lower rows are exact (along with the three columns)? It might not be a
complex, that is it might happen that gf \neq 0. Less widely known is
that if ker(g) \inc im(f), then it is also exact. That is actually if and
only if. That is, if either of K = ker(f) and I = im(f) contains the
other, they are equal. Well, I got to wondering about that and eventually
conjectured and proved that it is always the case that I/(I\cap K) is
isomorphic to K/(I\cap K). This makes it transparent that if either
contains the other, they have to be equal.
There is just one point remaining. I did this by chasing elements around
in Ab (so it is true in any abelian category). Does anyone see a clever
proof using the snake lemma? I don't.
Michael
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Date: Sat, 6 Sep 2008 12:48:14 +0200 (CEST)
From: Johannes Huebschmann
To: categories@mta.ca
Subject: categories: Categories and functors, query
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Dear All
I somewhat recall that, a while ago, we discussed the origins of
the notions of category and functor. S. Mac Lane had once pointed
out to me these origins but from my recollections we did not
entirely reproduce them.
In his paper
Samuel Eilenberg and Categories, JPAA 168 (2002), 127131
Saunders Mac Lane clearly pointed out the origins:
"Category" from Kant (which I had known all the time)
"Functor" from Carnap's book "Logical Syntax of Language" (which I
had forgotten).
Also I have a question, not directly related to the above issue:
I have seen, on some web page, a copy of
the referee's report about the EilenbergMac Lane paper
where EilenbergMac Lane spaces are introduced.
I cannot find this web page (or the report)
any more. Can anyone provide me with
a hint where I can possibly find it?
Many thanks in advance
Johannes
HUEBSCHMANN Johannes
Professeur de Mathematiques
USTL, UFR de Mathematiques
UMR 8524 Laboratoire Paul Painleve
F59 655 Villeneuve d'Ascq Cedex France
http://math.univlille1.fr/~huebschm
TEL. (33) 3 20 43 41 97
(33) 3 20 43 42 33 (secretariat)
(33) 3 20 43 48 50 (secretariat)
Fax (33) 3 20 43 43 02
email Johannes.Huebschmann@math.univlille1.fr
From rrosebru@mta.ca Sun Sep 7 20:00:55 2008 0300
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for categorieslist@mta.ca; Sun, 07 Sep 2008 19:55:25 0300
From: "R Brown"
To:
Subject: categories: Re: Categories and functors, query
Date: Sun, 7 Sep 2008 22:33:48 +0100
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There is another curiosity about the axioms for a category, namely the
infuence of the known axioms for a groupoid (Brandt, 1926). Bill Cockcroft
told me that these axioms had influenced EM. These axioms were well used in
the algebra group at Chicago. However when I asked Sammy about this in 1985
he firmly said `no, and was why the notion of groupoid did not appear as an
example in the EM paper'!
Perhaps it was a case of forgetting the influence?
Ronnie
 Original Message 
From: "Johannes Huebschmann"
To:
Sent: Saturday, September 06, 2008 11:48 AM
Subject: categories: Categories and functors, query
> Dear All
>
> I somewhat recall that, a while ago, we discussed the origins of
> the notions of category and functor. S. Mac Lane had once pointed
> out to me these origins but from my recollections we did not
> entirely reproduce them.
>
> In his paper
>
> Samuel Eilenberg and Categories, JPAA 168 (2002), 127131
>
> Saunders Mac Lane clearly pointed out the origins:
>
> "Category" from Kant (which I had known all the time)
>
> "Functor" from Carnap's book "Logical Syntax of Language" (which I
> had forgotten).
>
>
> Also I have a question, not directly related to the above issue:
>
> I have seen, on some web page, a copy of
> the referee's report about the EilenbergMac Lane paper
> where EilenbergMac Lane spaces are introduced.
> I cannot find this web page (or the report)
> any more. Can anyone provide me with
> a hint where I can possibly find it?
>
> Many thanks in advance
>
> Johannes
>
>
>
> HUEBSCHMANN Johannes
> Professeur de Mathematiques
> USTL, UFR de Mathematiques
> UMR 8524 Laboratoire Paul Painleve
> F59 655 Villeneuve d'Ascq Cedex France
> http://math.univlille1.fr/~huebschm
>
> TEL. (33) 3 20 43 41 97
> (33) 3 20 43 42 33 (secretariat)
> (33) 3 20 43 48 50 (secretariat)
> Fax (33) 3 20 43 43 02
>
> email Johannes.Huebschmann@math.univlille1.fr
>
>
>

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Subject: categories: Re: Categories and functors, query
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On Sep 7, 2008, at 2:33 PM, R Brown wrote:
> There is another curiosity about the axioms for a category, namely the
> infuence of the known axioms for a groupoid (Brandt, 1926). Bill
> Cockcroft
> told me that these axioms had influenced EM. These axioms were well
> used in
> the algebra group at Chicago. However when I asked Sammy about this
> in 1985
> he firmly said `no, and was why the notion of groupoid did not
> appear as an
> example in the EM paper'!
>
> Perhaps it was a case of forgetting the influence?
I certainly heard Saunders mention Brandt groupoids as examples.
(Not very good examples, since all maps are invertible.) But, as
everyone knows, it is not the definition of a category that
is the key part, but seeing that functors and natural
transformations are interesting.
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Date: Mon, 8 Sep 2008 08:50:41 0400 (EDT)
From: Michael Barr
To: categories@mta.ca
Subject: categories: Re: Categories and functors, query
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Interesting speculation, but how can we verify or refute it? What I can
add is that when I sat in on Sammy's category theory course (called
homological algebra, but I am not sure Ext or Tor were ever mentioned), I
do not recall that he so much as mentioned groupoids. I once mentioned to
Charles Ehresmann that he appeared to view categories as a generalization
of groupoids while Eilenberg and Mac Lane thought of them as a
generalization of posets. Charles agreed.
This reminds me of a speculation I have often had (although Saunders
denied and he knew Birkhoff pretty well). In the 30s and 40s, the word
"homomorphism" was regularly used but always meant surjective. By the
late 40s and 50s people were talking about "homomorphism into" meaning not
necessarily surjective. So groups had lattices of subgroups and lattices
of quotient groups and Birkhoff invented lattice theory at least partly in
the hope that the structure of those two lattices would tell you a lot
about the structure of the group. I don't think this actually happened to
any great extent. But I have wondered whether Birkhoff might instead have
invented categories had our more general notion of homomorphism been
rampant. As I said Saunders didn't think so, but it still sounds
attractive to me.
One of the things that astonishes me about "General theory of natural
equivalences" is that they clearly knew about natural transformations in
general but chose to talk only about equivalences. I once asked Sammy
about that and he more or less said something like one generalization at a
time. But they must have realized that the Hurevic map is a superior
example. Still, Steenrod must have gotten the point immediately.
Michael
On Sun, 7 Sep 2008, R Brown wrote:
> There is another curiosity about the axioms for a category, namely the
> infuence of the known axioms for a groupoid (Brandt, 1926). Bill Cockcroft
> told me that these axioms had influenced EM. These axioms were well used in
> the algebra group at Chicago. However when I asked Sammy about this in 1985
> he firmly said `no, and was why the notion of groupoid did not appear as an
> example in the EM paper'!
>
> Perhaps it was a case of forgetting the influence?
>
> Ronnie
>
>
>
>
>  Original Message 
> From: "Johannes Huebschmann"
> To:
> Sent: Saturday, September 06, 2008 11:48 AM
> Subject: categories: Categories and functors, query
>
>
>> Dear All
>>
>> I somewhat recall that, a while ago, we discussed the origins of
>> the notions of category and functor. S. Mac Lane had once pointed
>> out to me these origins but from my recollections we did not
>> entirely reproduce them.
>>
>> In his paper
>>
>> Samuel Eilenberg and Categories, JPAA 168 (2002), 127131
>>
>> Saunders Mac Lane clearly pointed out the origins:
>>
>> "Category" from Kant (which I had known all the time)
>>
>> "Functor" from Carnap's book "Logical Syntax of Language" (which I
>> had forgotten).
>>
>>
>> Also I have a question, not directly related to the above issue:
>>
>> I have seen, on some web page, a copy of
>> the referee's report about the EilenbergMac Lane paper
>> where EilenbergMac Lane spaces are introduced.
>> I cannot find this web page (or the report)
>> any more. Can anyone provide me with
>> a hint where I can possibly find it?
>>
>> Many thanks in advance
>>
>> Johannes
>>
>>
>>
>> HUEBSCHMANN Johannes
>> Professeur de Mathematiques
>> USTL, UFR de Mathematiques
>> UMR 8524 Laboratoire Paul Painleve
>> F59 655 Villeneuve d'Ascq Cedex France
>> http://math.univlille1.fr/~huebschm
>>
>> TEL. (33) 3 20 43 41 97
>> (33) 3 20 43 42 33 (secretariat)
>> (33) 3 20 43 48 50 (secretariat)
>> Fax (33) 3 20 43 43 02
>>
>> email Johannes.Huebschmann@math.univlille1.fr
>>
>>
>>
>
>
> 
>
>
>
> No virus found in this incoming message.
> Checked by AVG  http://www.avg.com
> Version: 8.0.169 / Virus Database: 270.6.17/1657  Release Date: 06/09/2008
> 20:07
>
>
>
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Date: Mon, 08 Sep 2008 12:00:58 0400
From: Walter Tholen
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There is another aspect to the EM achievement that I stressed in my
CT06 talk for the Eilenberg  Mac Lane Session at White Point. Given the
extent to which 20thcentury mathematics was entrenched in set theory,
it was a tremendous psychological step to put structure on "classes" and
to dare regarding these (perceived) monsters as objects that one could
study just as one would study individual groups or topological spaces.
In my experience, skepticism towards category theory is often rooted in
the fear of the "illegitimately large" size, till today. By comparison,
Brandt groupoids lived in the cozy and familiar small world, and their
definition was arrived at without having to leave the universe. With the
definition of category (and functor and natural transformation)
Eilenberg and Moore had to do a lot more than just repeating at the
monoid level what Brandt did at the group level! In my view their big
psychological step here is comparable to Cantor's daring to think that
there could be different levels of infinity.
Cheers,
Walter.
Dana Scott wrote:
>
> On Sep 7, 2008, at 2:33 PM, R Brown wrote:
>
>> There is another curiosity about the axioms for a category, namely the
>> infuence of the known axioms for a groupoid (Brandt, 1926). Bill
>> Cockcroft
>> told me that these axioms had influenced EM. These axioms were well
>> used in
>> the algebra group at Chicago. However when I asked Sammy about this
>> in 1985
>> he firmly said `no, and was why the notion of groupoid did not
>> appear as an
>> example in the EM paper'!
>>
>> Perhaps it was a case of forgetting the influence?
>
>
> I certainly heard Saunders mention Brandt groupoids as examples.
> (Not very good examples, since all maps are invertible.) But, as
> everyone knows, it is not the definition of a category that
> is the key part, but seeing that functors and natural
> transformations are interesting.
>
>
From rrosebru@mta.ca Mon Sep 8 21:15:52 2008 0300
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Date: Mon, 8 Sep 2008 15:04:00 0400
Subject: categories: Re: KT Chen's smooth CCC
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[Note from moderator: with apologies to the poster, this is being resent
since some of you will have received a version with some corrupted
characters.]
There are no
> objections to continuing to develop Chen's theory of "differentiable
> spaces"
Indeed on 8/17, 8/26, and 8/27 I urged the continuation of the development
of Chen's theory (for example the smooth space of piecewise smooth paths),
making use of recent experience of the range of possible categories.
It is possible that
>sheaves on a concrete site can never serve as a framework
> for differential geometry with infinitesimals.
But a proof would require a definition of what is meant by infinitesimals,
as well as the constraint on the framework that the maps 1>R and R>R
are the standard ones. Otherwise nonstandard analysis might fit.
The nilpotents or germs capture the Heraclitian nature of motion in a way
that abstract sets do not directly.
The misplaced concreteness, according to which
>spaces are [single] sets equipped with extra structure
is only a "second aspect of the default paradigm. The first aspect,
successfully overcome by the named pioneers, is the one generalizing the
default category of topological spaces (or locales). Here "default" refers
to the habitual response to the frequently occurring need to specify a
background category of cohesion in which to interpret our algebra.
The generalization from Sierpinskivalued functions (open sets) to
realvalued, has also been proposed, but that sort of
attempt never succeeded in yielding a simple theory of map spaces.
In contrast to this "functionalgebra X/R as primary" paradigm,
the semidual "figuregeometry S/X as primary" has led to good map
spaces (including internal function algebras) for many authors
(Sebastiao e Silva, Fox, Hurewicz 60 years ago and several more recent).
I believe that attempting to force nearlyperfect duality has in general
not led to good results, but of course one studies the extent to which
a monad (presumed identity on models S) approximates the identity on
general spaces. For example, Froelicher=E2=80=99s duality condition applies not
only to the line R but to the function space R^R, a nontrivial fact about
the smooth case, derived by LSZ from a study of distributions of compact
support (so citing it is not just namedropping).
Bill
On Tue 09/02/08 6:00 PM , John Baez baez@math.ucr.edu sent:
> Bill Lawvere wrote:
>=20
> >By urging the study of the good geometrical ideas
> and constructions of>Chen and Froelicher, as well as of Bott, Brown,
> Hurewicz, Mostow, Spanier,>Steenrod, I am of course not advocating the
> preferential resurrection of>the particular categories they tentatively
> devised to contain the>constructions.
>=20
> I chose Chen's framework when Urs Schreiber and I were doing some work
> in mathematical physics and we needed a "convenient category" of
> smoothspaces. I decided to choose one that was easy to explain to people
> brainwashed by the "default paradigm", in which spaces are sets
> equippedwith extra structure. Later I realized I needed to write a paper
> establishing some properties of Chen's framework. By doing that I
> guessI'm guilty of reinforcing the default paradigm, and for that I
> apologize.
> If I understand correctly, one can actually separate the objections
> to continuing to develop Chen's theory of "differentiable
> spaces"into two layers.
>=20
> Let me remind everyone of Chen's 1977 definition. He didn't state
> it this way, but it's equivalent:
>=20
> There's a category S whose objects are convex subsets C of R^n
> (n =3D 0,1,2,...) and whose maps are smooth maps between these.
> This category admits a Grothendieck pretopology where a cover
> is an open cover in the usual sense.
>=20
> A differentiable spaces is then a sheaf X on S. We think of
> X as a smooth space, and X(C) as the set of smooth maps from C to X.
>=20
> But the way Chen sets it up, differentiable spaces are not all
> the sheaves on S: just the "concrete" ones.
>=20
> These are defined using the terminal object 1 in S. Any convex set
> C has an underlying set of points hom(1,C). Any sheaf X on S has an
> underlying set of points X(1). Thanks to these, any element of X(C)
> has an underlying function from hom(1,C) to X(1). We say X is
> "concrete"if for all C, the map sending elements of X(C) to their underly=
ing
> functions is 11.
>=20
> The supposed advantage of concrete sheaves is that the underlying
> set functor X > X(1) is faithful on these. So, we can think of
> them as sets with extra structure.
>=20
> But this advantage is largely illusory. The concreteness condition
> is not very important in practice, and the concrete sheaves form not
> a topos, but only a quasitopos.
>=20
> That's one layer of objections. Of course, *these* objections
> can be answered by working with the topos of *all* sheaves on S.
> This topos contains some useful nonconcrete objects: for example,
> an object F such that F^X is the 1forms on X.
>=20
> But now comes a second layer of objections. This topos of sheaves
> still lacks other key features of synthetic differential geometry.
> Most importantly, it lacks the "infinitesimal arrow" object D
> suchthat X^D is the tangent bundle of X.
>=20
> The problem is that all the objects of S are ordinary
> "noninfinitesimal"spaces. There should only be one smooth map from any =
such space to D.
> So as a sheaf on S, D would be indistinguishable from the 1point
> space.
> So I guess the real problem is that the site S is concrete: that is,
> the functor assigning to any convex set C its set of points hom(1,C)
> is faithful. I could be jumping to conclusions, but it seems to me
> that that sheaves on a concrete site can never serve as a framework
> for differential geometry with infinitesimals.
>=20
> Best,
> jb
>=20
>=20
>=20
>=20
>=20
>=20
>=20
>=20
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for categorieslist@mta.ca; Mon, 08 Sep 2008 22:42:00 0300
Date: Mon, 8 Sep 2008 20:55:48 0400
From: tholen@mathstat.yorku.ca
To: categories@mta.ca
Subject: categories: Re: Categories and functors, query
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Quoting Walter Tholen :
> definition was arrived at without having to leave the universe. With the
> definition of category (and functor and natural transformation)
> Eilenberg and Moore had to do a lot more than just repeating at the
> monoid level what Brandt did at the group level! In my view their big
OOPS  "Moore" should read "Mac Lane", of course. (Sorry, Saunders!) W.
From rrosebru@mta.ca Tue Sep 9 09:02:02 2008 0300
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Date: Tue, 9 Sep 2008 14:53:03 +0400
From: "Nikita Danilov"
Subject: categories: Re: Categories and functors, query
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Michael Barr writes:
>
> This reminds me of a speculation I have often had (although Saunders
> denied and he knew Birkhoff pretty well). In the 30s and 40s, the word
> "homomorphism" was regularly used but always meant surjective. By the
> late 40s and 50s people were talking about "homomorphism into" meaning
not
> necessarily surjective. So groups had lattices of subgroups and lattices
> of quotient groups and Birkhoff invented lattice theory at least partly
in
> the hope that the structure of those two lattices would tell you a lot
> about the structure of the group. I don't think this actually happened
to
> any great extent. But I have wondered whether Birkhoff might instead
have
Noether's `set theoretic foundations of group theory', where group
axioms are based on a notion of coset decomposition rather than
multiplication, seems to be much earlier (20s) attempt to the same:
http://www.math.jussieu.fr/~leila/grothendieckcircle/mclarty2.pdf
>
> Michael
Nikita.
From rrosebru@mta.ca Tue Sep 9 19:18:27 2008 0300
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Date: Tue, 09 Sep 2008 18:05:23 0400
From: jim stasheff
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Michael
But they must have realized that the Hurevic map is a superior
example. Still, Steenrod must have gotten the point immediately.
You lost me there.
jim
From rrosebru@mta.ca Wed Sep 10 12:08:20 2008 0300
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Date: Tue, 09 Sep 2008 18:22:10 0400
From: jim stasheff
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Walter,
I beg to differ only with
In my experience, skepticism towards category theory is often rooted in
the fear of the "illegitimately large" size, till today.
In my experience, disdain for cat theory is due to papers with a very
high density of unfamiliar names
reminiscent of the minutia of PST and the (in) famous comment (by some
one) about something like:
hereditary hemidemisemigroups with chain condition
jim
Tholen wrote:
> There is another aspect to the EM achievement that I stressed in my
> CT06 talk for the Eilenberg  Mac Lane Session at White Point. Given the
> extent to which 20thcentury mathematics was entrenched in set theory,
> it was a tremendous psychological step to put structure on "classes" and
> to dare regarding these (perceived) monsters as objects that one could
> study just as one would study individual groups or topological spaces.
> In my experience, skepticism towards category theory is often rooted in
> the fear of the "illegitimately large" size, till today. By comparison,
> Brandt groupoids lived in the cozy and familiar small world, and their
> definition was arrived at without having to leave the universe. With the
> definition of category (and functor and natural transformation)
> Eilenberg and Moore had to do a lot more than just repeating at the
> monoid level what Brandt did at the group level! In my view their big
> psychological step here is comparable to Cantor's daring to think that
> there could be different levels of infinity.
>
> Cheers,
> Walter.
From rrosebru@mta.ca Wed Sep 10 13:33:51 2008 0300
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for categorieslist@mta.ca; Wed, 10 Sep 2008 13:25:51 0300
Date: Tue, 9 Sep 2008 20:17:14 0700
From: "Alex Hoffnung"
To: categories@mta.ca
Subject: categories: Equivalence of pseudolimits
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Hi all,
Given an indexing 2category J, a pair of parallel functors
F,G : J > CAT, and a natural equivalence f : F ==> G,
the pseudolimits of F and G should be equivalent.
I am trying to find out what paper, if any, I can cite for this theorem. Or
maybe this is just the type of thing that nobody has bothered to write down.
Any help would be appreciated.
best,
Alex Hoffnung
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Subject: categories: The disdain for categories
Date: Wed, 10 Sep 2008 13:35:22 0400
From: Andre Joyal
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Jim Stasheff wrote:
>In my experience, disdain for cat theory is due to papers with a very
>high density of unfamiliar names
>reminiscent of the minutia of PST and the (in) famous comment (by some
>one) about something like:
>hereditary hemidemisemigroups with chain condition
The chosen example is not too convincing,=20
since the notions involved are not typically categorical.
Complicated sentences like this can be found in every fields.
They are often the mark of a poor paper.
My guess is that the disdain for categories has a mixed origin.
Like logic, category theory has a taste for generalities.
But most mathematicians are specialised and they find
it hard to believe that important progresses can be made in their fields
from the outside, as the result of general insights.
But we all know that the division of mathematics into fields=20
is justified more by sociology than by science.=20
Category theory is a powerful tool for crossing=20
the boundaries between the fields.
The unity of mathematics is growing stronger every day.
andre
 Message d'origine
De: catdist@mta.ca de la part de jim stasheff
Date: mar. 09/09/2008 18:22
=C0: categories@mta.ca
Objet : categories: Re: Categories and functors, query
=20
Walter,
I beg to differ only with
In my experience, skepticism towards category theory is often rooted in
the fear of the "illegitimately large" size, till today.
In my experience, disdain for cat theory is due to papers with a very
high density of unfamiliar names
reminiscent of the minutia of PST and the (in) famous comment (by some
one) about something like:
hereditary hemidemisemigroups with chain condition
jim
Tholen wrote:
> There is another aspect to the EM achievement that I stressed in my
> CT06 talk for the Eilenberg  Mac Lane Session at White Point. Given =
the
> extent to which 20thcentury mathematics was entrenched in set theory,
> it was a tremendous psychological step to put structure on "classes" =
and
> to dare regarding these (perceived) monsters as objects that one could
> study just as one would study individual groups or topological spaces.
> In my experience, skepticism towards category theory is often rooted =
in
> the fear of the "illegitimately large" size, till today. By =
comparison,
> Brandt groupoids lived in the cozy and familiar small world, and their
> definition was arrived at without having to leave the universe. With =
the
> definition of category (and functor and natural transformation)
> Eilenberg and Moore had to do a lot more than just repeating at the
> monoid level what Brandt did at the group level! In my view their big
> psychological step here is comparable to Cantor's daring to think that
> there could be different levels of infinity.
>
> Cheers,
> Walter.
From rrosebru@mta.ca Thu Sep 11 14:19:57 2008 0300
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for categorieslist@mta.ca; Thu, 11 Sep 2008 14:12:22 0300
Date: Wed, 10 Sep 2008 18:12:10 0500
From: "Charles Wells"
To: catbb
Subject: categories: New version of Graph Based Logic and Sketches
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I have posted a new version of Graph Based Logic and Sketches, by Atish
Bagchi and Charles Wells, here:
http://www.cwru.edu/artsci/math/wells/pub/pdf/gbls.pdf
This will eventually appear on ArXiv.

Charles Wells
professional website: http://www.cwru.edu/artsci/math/wells/home.html
blog: http://www.gyregimble.blogspot.com/
abstract math website: http://www.abstractmath.org/MM//MMIntro.htm
personal website: http://www.abstractmath.org/Personal/index.html
From rrosebru@mta.ca Thu Sep 11 14:20:13 2008 0300
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for categorieslist@mta.ca; Thu, 11 Sep 2008 14:13:33 0300
Date: Wed, 10 Sep 2008 17:20:45 0700
From: Toby Bartels
To: categories@mta.ca
Subject: categories: Re: Categories and functors, query
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Dana Scott wrote in part:
>But, as
>everyone knows, it is not the definition of a category that
>is the key part, but seeing that functors and natural
>transformations are interesting.
Indeed, the notion of natural isomorphism (or canonical isomorphism)
should be available already to groupoid theorists before 1945.
To what extent did they know about functors and natural isomorphisms,
and to what extent did Saunders & Mac Lane have to tell them?
Or, pace Walter's remarks, did they know about the ~small~ ones
but not have the guts to apply them to large classes of strucures?
It's been said before that the real insight of category theory
as something more general than groupoids, monoids, and posets
is the notion of adjoint functors (including limits, etc).
I'm inclined to agree, so I'm interested in why and whether
groupoid theorists thought of (and applied) that which they ~did~ have.
Toby
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for categorieslist@mta.ca; Thu, 11 Sep 2008 14:25:20 0300
Date: Wed, 10 Sep 2008 21:25:24 0400
From: jim stasheff
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Andre Joyal wrote:
> Jim Stasheff wrote:
>
> =20
>> In my experience, disdain for cat theory is due to papers with a very
>> high density of unfamiliar names
>> reminiscent of the minutia of PST and the (in) famous comment (by some
>> one) about something like:
>> hereditary hemidemisemigroups with chain condition
>> =20
>
> The chosen example is not too convincing,=20
> since the notions involved are not typically categorical.
> Complicated sentences like this can be found in every fields.
> They are often the mark of a poor paper.
> Category theory is a powerful tool for crossing=20
> the boundaries between the fields.
> The unity of mathematics is growing stronger every day.
>
> andre
>
> =20
Indeed, the quote I was misremembering was NOT in category theory
how's that for crossing the boundaries between the fields. ;)
in fact, it turns out that the correct usage is
hemidemisemiquaver  in music!
jim
>  Message d'origine
> De: catdist@mta.ca de la part de jim stasheff
> Date: mar. 09/09/2008 18:22
> =C0: categories@mta.ca
> Objet : categories: Re: Categories and functors, query
> =20
> Walter,
>
> I beg to differ only with
>
> In my experience, skepticism towards category theory is often rooted in
> the fear of the "illegitimately large" size, till today.
>
> In my experience, disdain for cat theory is due to papers with a very
> high density of unfamiliar names
> reminiscent of the minutia of PST and the (in) famous comment (by some
> one) about something like:
> hereditary hemidemisemigroups with chain condition
>
> jim
> Tholen wrote:
> =20
>> There is another aspect to the EM achievement that I stressed in my
>> CT06 talk for the Eilenberg  Mac Lane Session at White Point. Given t=
he
>> extent to which 20thcentury mathematics was entrenched in set theory,
>> it was a tremendous psychological step to put structure on "classes" a=
nd
>> to dare regarding these (perceived) monsters as objects that one could
>> study just as one would study individual groups or topological spaces.
>> In my experience, skepticism towards category theory is often rooted i=
n
>> the fear of the "illegitimately large" size, till today. By comparison=
,
>> Brandt groupoids lived in the cozy and familiar small world, and their
>> definition was arrived at without having to leave the universe. With t=
he
>> definition of category (and functor and natural transformation)
>> Eilenberg and Moore had to do a lot more than just repeating at the
>> monoid level what Brandt did at the group level! In my view their big
>> psychological step here is comparable to Cantor's daring to think that
>> there could be different levels of infinity.
>>
>> Cheers,
>> Walter.
>> =20
>
>
>
>
> =20
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for categorieslist@mta.ca; Thu, 11 Sep 2008 14:26:24 0300
Date: Thu, 11 Sep 2008 12:27:33 +1000
From: "Dominic Verity"
To: categories@mta.ca
Subject: categories: Re: Equivalence of pseudolimits
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Hi Alex,
This property is certainly known to hold for a much larger class of
2categorical limits  the flexible limits, which class includes the classes
of pseudo, lax and oplax limits. I believe you will find a proof of this
result in the Bird, Kelly, Power and Street paper entitled "Flexible Limits
for 2Categories". Failing that the Power and Robinson paper "A
characterisation of PIE limits" probably contains this result is some form.
You can also find explications of the pseudo and lax results in earlier
works by Street, although obvious candidates for the best one to consult
elude me at the moment.
The most elementary proof of the flexible limit result starts by observing
that all flexible limits can be constructed using products, splittings of
idempotents and a couple of less familiar, exclusively 2categorical, limits
called inserters and equifiers. It is then straight forward to verify the
result you mention for each of these particular limits and then to infer
that it must therefore hold for all flexible limits.
In the early 1990's Robert Pare introduced a class of limits called the
persistent limits. These were defined for 2categories, but made use of his
double categorical approach to 2limits. Persistent limits are precisely
those limits which have the stability property you seek, but with regard to
a slightly more general class of double categorical diagram transformations
whose 1cellular components are all equivalences.
In my thesis (1992), I prove that the class of flexible limits introduced by
Bird, Kelly, Power and Street is identical to Pare's class of persistent
limits  thus closing the circle and demonstrating that the flexible limits
are in a natural sense the largest class of 2limits which have this
property.
Regards
Dominic Verity
2008/9/10 Alex Hoffnung
> Hi all,
>
> Given an indexing 2category J, a pair of parallel functors
> F,G : J > CAT, and a natural equivalence f : F ==> G,
> the pseudolimits of F and G should be equivalent.
>
> I am trying to find out what paper, if any, I can cite for this theorem.
> Or
> maybe this is just the type of thing that nobody has bothered to write
> down.
> Any help would be appreciated.
>
> best,
> Alex Hoffnung
>
From rrosebru@mta.ca Thu Sep 11 14:34:43 2008 0300
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for categorieslist@mta.ca; Thu, 11 Sep 2008 14:27:22 0300
To: categories@mta.ca
From: "Edward A. Hirsch"
Subject: categories: CSR2009: First Call for Papers
Date: Thu, 11 Sep 2008 10:51:27 +0400 (MSD)
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*************** We apologize for multiple copies ******************************
First Call for Papers
4th INTERNATIONAL COMPUTER SCIENCE SYMPOSIUM IN RUSSIA (CSR 2009)
August 1823, 2009, Novosibirsk, Russia
http://math.nsc.ru/conference/csr2009/index.shtml
*******************************************************************************
CSR 2009 intends to reflect the broad scope of international cooperation in
computer science. It is the 4th conference in a series of regular events
started with CSR 2006 in St.Petersburg (see LNCS 3967), CSR 2007 in
Ekaterinburg (see LNCS 4649), and CSR 2008 in Moscow (see LNCS 5010). As usual,
CSR 2009 consists of two tracks: Theory Track and Applications and Technology
Track.
IMPORTANT DATES:
Deadline for submissions: November 26, 2008
Notification of acceptance: February 3, 2009
Conference dates: August 1823, 2009
TOPICS
Theory Track topics include
* algorithms and data structures
* complexity and cryptography
* formal languages and automata
* computational models and concepts
* proof theory and applications of logic to computer science.
Application Track topics include
* abstract interpretation
* model checking
* automated reasoning
* deductive methods
* constraint solving
* functional and declarative languages
* type systems
* software engineering
* development methodologies
for design, development, testing, analysis, and verification of correct
and reliable systems.
OPENING LECTURE:
Andrei Voronkov (University of Manchester).
PROGRAM COMMITTEES
Program committee of Theory Track is:
Farid Ablayev, Kazan State University
Sergei N. Artemov, City University of New York
Lev Beklemishev, Steklov Inst., Moscow
Veronique Bruyere, Universite de MonsHainaut
Cristian Calude, University of Auckland
Christian Glasser, Universitaet Wuerzburg
Dima Grigoriev, Institut de Recherche Mathematique de Rennes
Dietrich Kuske, Universitaet Leipzig
Larisa Maksimova, IM, Novosibirsk
Andrei Mantsivoda, Irkutsk State University
Yuri Matiyasevich, Steklov Institute, St.Petersburg
Elvira Mayordomo, Universidad de Zaragoza
Pierre McKenzie, Universite de Montreal
Andrey S. Morozov, IM, Novosibirsk (cochair)
JeanEric Pin, LIAFA, Paris
Kai Salomaa, Queen's University, Kingston, Canada
Victor Selivanov, Novosibirsk Pedagogical University
Ludwig Staiger, Universitaet HalleWittenberg
Klaus W. Wagner, Universitaet Wuerzburg (cochair)
Program committee of Applications and Technology Track is:
Thomas Ball, Microsoft Research
Josh Berdine, Microsoft Research
Bart Demoen, K.U. Leuven
Franjo Ivancic, NEC Laboratories America
Martin Leucker, TU Munich
Rupak Majumdar, University of California, Los Angeles
Greg Morrisett, Harvard University
Arnd PoetzschHeffter, University of Kaiserslautern
Andreas Rossberg, MPISWS
Andrey Rybalchenko, MPISWS (chair)
Alexander Serebrenik, TU Eindhoven
Henny Sipma, Stanford University
Natasha Sharygina, University of Lugano
Helmut Veith, TU Darmstadt
Eran Yahav, IBM Research
Andreas Zeller, Universitaet des Saarlandes
ORGANIZERS:
Sobolev Institute of Mathematics SB RAS.
Conference chair: Anna Frid
SUBMISSIONS:
Authors are invited to submit an extended abstract or a full paper of at most
10 pages in the LNCS format (the instructions on it can be found here:
http://www.springer.com/computer/lncs?SGWID=01647723760). Proofs and
other material omitted due to space constraints are to be put into a clearly
marked appendix to be read at discretion of the referees. Papers must present
original (and not previously published) research. Simultaneous submissions
to journals or to other conferences with published proceedings are not
allowed. The proceedings of the symposium will be published in Springer's
LNCS series.
Submissions should be uploaded at EasyChair Conference system:
http://www.easychair.org/conferences/?conf=csr09 .
FURTHER INFORMATION AND CONTACTS:
Web: http://math.nsc.ru/conference/csr2009/index.shtml
Email: csr2009@math.nsc.ru
From rrosebru@mta.ca Thu Sep 11 14:38:31 2008 0300
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for categorieslist@mta.ca; Thu, 11 Sep 2008 14:31:56 0300
From: "R Brown"
To:
Subject: categories: Re: Categories and functors, query
Date: Thu, 11 Sep 2008 10:05:10 +0100
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Dear All,
I agree with Andre that part of the matter is sociological. It is also quite
fundamental, and is about the proper aims of mathematics. The need is for
discussion, rather than total agreement.
Miles Reid's infamous comment was "The study of category theory for its own
sake (surely one of the most sterile of intellectual pursuits) also
dates from this time; Grothendieck can't necessarily be blamed
for this, [!!!] since his own use of categories was very successful in
solving problems. " (My riposte in a paper was to suggest a game: `I can
think of a more intellectually sterile pursuit than you can!') This suggests
the view
that solving problems, presumably already formulated ones, is the key part
of mathematics. (Miles did tell me he expected his student to use topoi or
whatever!) A 1974 report on graduate mathematicians in employment suggested
they were good at solving problems but not so good at formulating them.
Grothendieck in one letter to me wrote on his aim for
`understanding'. (see my article on `Promoting Mathematics' on my
Popularisation web page) I believe many students come into mathematics
because they like finding out why things are true, they want to understand.
Loday told me he thought one of the strengths of French mathematics was to
try to realise this aim. By contrast, I once
asked Frank Adams why he wrote that a certain nonabelian cohomology was
trivial and he said `you just do a calculation'  Frank was a determined
problem solver!
So people have asked: "Where are the big theorems, the big problems, in
category theory?" Are they there? Does it matter if they are not there?
Atiyah in his article on `20th century mathematics' (Bull LMS, 2001) talks
about the unity of mathematics, but the word `category' does not occur in
his
article. (Neither does groupoid.) He states a dichotomy between geometry
(good)and algebra (bad) but fails to recognise the combination given by,
say, Grothendieck's work, and also by higher categorical structures.
Indeed, underlying structures and processes may be of various types, all
very useful to know. I am *very* impressed by Henry Whitehead's finding so
many of these.
A word often omitted in mathematics teaching is `analogy'. Yet this is what
abstraction is about, and why it is so powerful. Category theory allows for
powerful analogies.
I am always puzzled, even horrified, by mathematicians who use the word
`nonsense' to describe the work of others (as is all too common), yet often
themselves cannot well define professionalism in the subject. Indeed they
often cannot believe the direction others may take is chosen for good
professional reasons! They sometimes say `not mainstream'. Yet history shows
`the mainstream' shifts its course radically over the years. The lack is of
a consistent and well maintained mathematical criticism, recognising
historical trends and not just the `great man (or woman)', or famous
problem, approach.
I believe we need to have prepared an answer to: What has category theory
done for mathematics? And indeed for evaluation of any subject areas. But a
good case is that category theory leads, or can lead, and has led, to
clarity, to understanding and development of the rich variety of structures
there are and to be found. However this does not rate for million $ prizes
(as it should, of course!).
When I see all the current fuss (rightly) about the LHC in Geneva, I do
wonder: who is going to speak up for mathematics, to attract students into
the subject, by getting over a message as to its value and achievements? and
also getting this message over to students studying the subject! (see
`Promoting Mathematics' and Tim and my article on `the methodology of
mathematics')
Ronnie
www.bangor.ac.uk/r.brown/publar.html
 Original Message 
From: "jim stasheff"
To:
Sent: Tuesday, September 09, 2008 11:22 PM
Subject: categories: Re: Categories and functors, query
> Walter,
>
> I beg to differ only with
>
> In my experience, skepticism towards category theory is often rooted in
> the fear of the "illegitimately large" size, till today.
>
> In my experience, disdain for cat theory is due to papers with a very
> high density of unfamiliar names
> reminiscent of the minutia of PST and the (in) famous comment (by some
> one) about something like:
> hereditary hemidemisemigroups with chain condition
>
> jim
> Tholen wrote:
>> There is another aspect to the EM achievement that I stressed in my
>> CT06 talk for the Eilenberg  Mac Lane Session at White Point. Given the
>> extent to which 20thcentury mathematics was entrenched in set theory,
>> it was a tremendous psychological step to put structure on "classes" and
>> to dare regarding these (perceived) monsters as objects that one could
>> study just as one would study individual groups or topological spaces.
>> In my experience, skepticism towards category theory is often rooted in
>> the fear of the "illegitimately large" size, till today. By comparison,
>> Brandt groupoids lived in the cozy and familiar small world, and their
>> definition was arrived at without having to leave the universe. With the
>> definition of category (and functor and natural transformation)
>> Eilenberg and Moore had to do a lot more than just repeating at the
>> monoid level what Brandt did at the group level! In my view their big
>> psychological step here is comparable to Cantor's daring to think that
>> there could be different levels of infinity.
>>
>> Cheers,
>> Walter.
>
From rrosebru@mta.ca Fri Sep 12 11:42:09 2008 0300
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for categorieslist@mta.ca; Fri, 12 Sep 2008 11:35:59 0300
Date: Thu, 11 Sep 2008 18:54:11 +0100 (BST)
From: Richard Garner
To: categories@mta.ca
Subject: categories: Re: Equivalence of pseudolimits
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In the case singled out by Alex a rather direct proof can
also be given. Let Psd denote the 2category of 2functors,
pseudonatural transformations and modifications P > Cat. The
pseudolimit of a 2functor F: P > Cat may be identified with
the homcategory Psd(1,F); and accordingly we have a
pseudolimit 2functor lim = Psd(1,): Psd > Cat, which sends
pseudonatural equivalences F =~ G to equivalences of
categories lim(F) =~ lim(G). The corresponding result for
pseudolimits in other 2categories now follows by the
Catenriched Yoneda lemma.
Richard
On 11 September 2008 12:27 Dominic Verity wrote:
> Hi Alex,
>
> This property is certainly known to hold for a much larger class of
> 2categorical limits  the flexible limits, which class includes the classes
> of pseudo, lax and oplax limits. I believe you will find a proof of this
> result in the Bird, Kelly, Power and Street paper entitled "Flexible Limits
> for 2Categories". Failing that the Power and Robinson paper "A
> characterisation of PIE limits" probably contains this result is some form.
>
> You can also find explications of the pseudo and lax results in earlier
> works by Street, although obvious candidates for the best one to consult
> elude me at the moment.
>
> The most elementary proof of the flexible limit result starts by observing
> that all flexible limits can be constructed using products, splittings of
> idempotents and a couple of less familiar, exclusively 2categorical, limits
> called inserters and equifiers. It is then straight forward to verify the
> result you mention for each of these particular limits and then to infer
> that it must therefore hold for all flexible limits.
>
> In the early 1990's Robert Pare introduced a class of limits called the
> persistent limits. These were defined for 2categories, but made use of his
> double categorical approach to 2limits. Persistent limits are precisely
> those limits which have the stability property you seek, but with regard to
> a slightly more general class of double categorical diagram transformations
> whose 1cellular components are all equivalences.
>
> In my thesis (1992), I prove that the class of flexible limits introduced by
> Bird, Kelly, Power and Street is identical to Pare's class of persistent
> limits  thus closing the circle and demonstrating that the flexible limits
> are in a natural sense the largest class of 2limits which have this
> property.
>
> Regards
>
> Dominic Verity
>
> 2008/9/10 Alex Hoffnung
>
>> Hi all,
>>
>> Given an indexing 2category J, a pair of parallel functors
>> F,G : J > CAT, and a natural equivalence f : F ==> G,
>> the pseudolimits of F and G should be equivalent.
>>
>> I am trying to find out what paper, if any, I can cite for this theorem.
>> Or
>> maybe this is just the type of thing that nobody has bothered to write
>> down.
>> Any help would be appreciated.
>>
>> best,
>> Alex Hoffnung
>>
>
>
From rrosebru@mta.ca Fri Sep 12 11:43:01 2008 0300
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for categorieslist@mta.ca; Fri, 12 Sep 2008 11:37:14 0300
Date: Thu, 11 Sep 2008 17:12:37 0400
From: Walter Tholen
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Here is some uplifting 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),
pp128136. 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 nonFrench 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."
From rrosebru@mta.ca Fri Sep 12 11:44:39 2008 0300
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for categorieslist@mta.ca; Fri, 12 Sep 2008 11:38:58 0300
Date: Thu, 11 Sep 2008 19:01:57 0400
From: edubuc
To: categories@mta.ca
Subject: categories: categories and disdain
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this is about the recent thread "categories and functors" and "disdain
for categories"
Ten points:
1) It seems clear that EM arrived to categories and functors by
abstraction from the usual large categories of sets, groups, boolean
algebras, modules, etc etc
2) It seems (less clearly) that Ehresman arrived to categories and
functors by generalization from groupoids and morphisms of groupoids.
3) I agree with "It's been said before that the real insight of
category theory
as something more general than groupoids, monoids, and posets
is the notion of adjoint functors (including limits, etc)."
Concerning this, I think that real breakthrough made by categories is
the simple fact that they furnish the appropriate abstract structure to
define the Bourbaki's concept of universal property. The fact that the
singleton set is characterized by being a terminal object opens the way
to characterize thousands of objects and constructions by being the
terminal object in the appropriate category. Yoneda's lemma is the
milestone. Everything is due to it.
4) I think the smalllarge business played no role at all in the rise of
the concept of categories, neither in the rise of the disdain to them by
many mathematicians.
5) "working" mathematicians were never afraid about paradoxes. In
consequence, I think that phrases as "dare regarding these (perceived)
monsters ...", "fear of the "illegitimately large" size", etc etc,
are misleading and out of place.
6) Cantor did not "dare to think that there could be different levels
of infinity", he discovered that they were different levels of infinity,
and proceed to study this phenomena. This was not a bold action, he was
just fascinated by the existence of different levels of infinity. He was
not afraid of paradoxes either, he was very well aware of Russell
paradox, but for him it was just another theorem.
7) It is often repeated that axiomatic set theory arise in order to
eliminate paradoxes. False, axiomatic set theory arise in an attempt to
understand Von Neumann accumulation process: Which were the axioms
satisfied by the output of that process ? Answer: axiomatic set theory.
8) "In my experience, disdain for cat theory is due to papers with a very
high density of unfamiliar names", I agree with this in the sense that
this fact contributed to the rise of the disdain, but not as the single
reason. I agree also with "Complicated sentences like this can be found
in every fields. They are often the mark of a poor paper".
It follows there must be other reasons (besides the abundance of poor
papers in category theory, a fact that I found true) to explain the disdain.
9) A profound reason could be an instinctive opposition to real change
in many people. The instinctive reaction against progress that may
disrupt their own comfortable position.
10) The so self proclaimed "problem solvers" who disdain abstract
theories often do not resolve any real problem. The just do "concrete
nonsense". People who really solve true problems usually have a great
respect for abstract theories. Of course, they are also many who just
do "abstract nonsense" instead of contribute to the meaningful
development of theories.
eduardo dubuc
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Date: Fri, 12 Sep 2008 02:56:31 0700 (PDT)
From: Jeff Egger
Subject: categories: Another terminological question...
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Dear all,
In ``basic concepts of enriched category theory'',
Kelly writes:
> Since the conetype limits have no special position of
> dominancein the general case, we go so far as to call
> weighted limits simply ``limits'', where confusion
> seems unlikely.
My question is this: why does he not apply the same
principle to the concept of powers? Instead, he
introduces the word ``cotensor'', apparently in order
to reserve the word ``power'' for that special case
which could sensibly be called ``discrete power''.
[This leads to the unfortunate scenario that a
``cotensor'' is a sort of limit, while dually a
``tensor'' is a sort of colimit.] Is there perhaps
some genuinely mathematical objection to calling
cotensors powers (and tensors copowers) which I may
have overlooked?
Cheers,
Jeff.
P.S. I specify ``genuinely mathematical'' because I
know that some people are opposed to any change of
terminology for any reason whatsoever. Obviously,
I disagree; in particular, I don't see that minor
terminological schisms such as monad/triple (even
compact/rigid/autonomous) are in any way detrimental
to the subject.
I also disagree with the notion (symptomatic of the
curiously feudal mentality which seems to permeate the
mathematical community) that prestigious mathematicians
have more right to set terminology than the rest of us.
I see no correlation between mathematical talent and
good terminology; nor do I understand that a great
mathematician can be ``dishonoured'' by anything less
than strict adherence to their terminologyor notation,
for that matter.
__________________________________________________________________
Looking for the perfect gift? Give the gift of Flickr!
http://www.flickr.com/gift/
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for categorieslist@mta.ca; Fri, 12 Sep 2008 14:56:55 0300
Date: Fri, 12 Sep 2008 17:57:00 +0200
From: "zoran skoda"
Subject: categories: Re: Bourbaki and Categories
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>
> 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 nonacceptance 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 structureoriented 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 selfconscious (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
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From: Colin McLarty
To: categories@mta.ca
Date: Fri, 12 Sep 2008 14:46:11 0400
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From=3A zoran skoda =3Czskoda=40gmail=2Ecom=3E
Date=3A Friday=2C September 12=2C 2008 2=3A06 pm
wrote=2C among other things
=3E main points of departure=2E The remark that as a proponent of =
=3E =22structures=22 Bourbaki
=3E had to include categories is anyway a bit lacking an argument=2E =
=3E First of all=2C because
=3E of the size problems one can not take big categories on equal =
=3E footing with=2C say groups=2C
=3E and considering only small categories would be strange and lacking =
=3E most interesting examples=2E
The claim is not that Bourbaki should have studied categories as
structures=2E It is that Bourbaki was doomed to fail in trying to use
their structure theory=2E Leo Corry shows in his book =22Modern Algebra =
and
the Rise of Mathematical Structures=22 (Birkh=E4user 1996) that they did =
fail=2E =
And they should have seen this coming=2C because their theory had been =
=22superseded by that of category and functor=2C which includes it under =
a
more general
and convenient form=22 (Dieudonn=E9 =22The Work of Nicholas Bourbaki=22 1=
970)=2E
best=2C Colin
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Date: Fri, 12 Sep 2008 16:34:55 0400 (EDT)
From: Robert Seely
To: categories@mta.ca
Subject: categories: Re: Bourbaki and Categories
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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 =

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for categorieslist@mta.ca; Fri, 12 Sep 2008 21:10:55 0300
Date: Fri, 12 Sep 2008 23:05:56 +0200 (CEST)
Subject: categories: Bourbaki and categories (references)
From: pierre.ageron@math.unicaen.fr
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On the Bourbaki's attitude towards category theory, there are various
references (check online Liliane Beaulieu's bibliography on Bourbaki), bu=
t
the fundamental study is undoubtedly that of Ralf Kr=F6mer. A very very g=
ood
paper, limited only by the nonavailability of Bourbaki's archives for th=
e
sixties :
KR=D6MER, Ralf
La =AB machine de Grothendieck =BB se fondetelle seulement sur des voca=
bles
m=E9tamath=E9matiques? Bourbaki et les cat=E9gories au cours des ann=E9es
cinquante, Revue d'histoire des math=E9matiques 121 (2006), pages 11916=
2
Thank you to Paul for remembering my talk at Amiens. It was certainly
related to the subject and also relied on Bourbaki's archives, but dealt
with an earlier "precategories" period. Only a 2 pages abstract is so
far available (I might write down a longer version some day in sha'a
llah) :
AGERON, Pierre
Autour d=92Ehresmann : Bourbaki, Cavaill=E8s, Lautman, Cahiers de topolog=
ie et
de g=E9om=E9trie diff=E9rentielle cat=E9goriques XLVI3, pages 165166
(available online via NUMDAM)
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for categorieslist@mta.ca; Sat, 13 Sep 2008 19:07:15 0300
From: Colin McLarty
To: categories@mta.ca
Date: Fri, 12 Sep 2008 21:25:50 0400
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From: Robert Seely
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
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From: "George Janelidze"
To:
Subject: categories: Re: Bourbaki and Categories
Date: Sat, 13 Sep 2008 16:31:19 +0200
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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
muchmuchmuch 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
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Date: Sat, 13 Sep 2008 13:17:23 0400
From: Andre Joyal
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Dear Colin, Zoran, Robert, Eduardo and All,
I find the present discussion on Bourbaki and category theory very =
important. =20
I recall asking the question to Samuel Eilenberg 25 years ago and more =
recently to Pierre Cartier. =20
If my recollection is right, Bourbaki had essentially two options: =
rewrite the whole treaty using categories,=20
or just introduce them in the book on homological algebra,=20
The second option won, essentially because of the enormity of the task =
of rewriting everything.=20
Other factors may have contributed on a smaller scale, like some =
unresolved foundational questions.=20
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.=20
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. =20
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.=20
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.=20
Bourbaki is dead but I hope that the humanistic philosophy behind the =
enterprise is not. =20
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.=20
Millions of people need and want to learn science and mathematics.=20
Should we not try to give Bourbaki a second life?=20
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: catdist@mta.ca de la part de Colin McLarty
Date: ven. 12/09/2008 14:46
=C0: categories@mta.ca
Objet : categories: Re: Bourbaki and Categories
=20
From: zoran skoda
Date: Friday, September 12, 2008 2:06 pm
wrote, among other things
> main points of departure. The remark that as a proponent of=20
> "structures" Bourbaki
> had to include categories is anyway a bit lacking an argument.=20
> First of all, because
> of the size problems one can not take big categories on equal=20
> footing with, say groups,
> and considering only small categories would be strange and lacking=20
> 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" (Birkh=E4user 1996) that they did =
fail. =20
And they should have seen this coming, because their theory had been=20
"superseded by that of category and functor, which includes it under a
more general
and convenient form" (Dieudonn=E9 "The Work of Nicholas Bourbaki" 1970).
best, Colin
From rrosebru@mta.ca Sun Sep 14 13:58:59 2008 0300
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id 1Keuqh0006zbNs
for categorieslist@mta.ca; Sun, 14 Sep 2008 13:53:23 0300
From: "R Brown"
To:
Subject: categories: Re: Bourbaki and Categories
Date: Sun, 14 Sep 2008 11:24:18 +0100
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Dear All,
The importance of Bourbaki should be stessed, as it was started when, so =
we=20
are told, texts were very bad. There are many beautiful things in the boo=
ks:=20
I developed part of an undergraduate course from the account of the=20
classification of closed subgroups of R^n. This relates to old questions =
on=20
orbits of the planets, and also gives some nice exercises and even exam=20
questions of a calculation type. It is good to present students with a=20
classification theorem.
The difficulties for Bourbaki seem to arise from the presentation (a) as =
a=20
final and definitive view in toto, and (b) without enough context, as And=
re=20
points out.
On (a), there is the old childish joke: what happens if you put worms in =
a=20
straight line from Marble Arch to Picadilly Circus? One of them would be=20
bound to wriggle and spoil it all! So some mathematical worms have not on=
ly=20
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=20
authoritative, and will last indefinitely.
What Bourbaki also shows is the value for at least the writers of taking =
a=20
viewpoint and following it through as far as it will go: if it seems in t=
he=20
end to go too far, or to be inadequate, then that is valuable information=
=20
for them and others. See my Dirac quote in `Out of Line'.
Ronnie
 Original Message =20
From: "Andre Joyal"
To:
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=20
important.
I recall asking the question to Samuel Eilenberg 25 years ago and more=20
recently to Pierre Cartier.
If my recollection is right, Bourbaki had essentially two options: rewrit=
e=20
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=
=20
rewriting everything.
Other factors may have contributed on a smaller scale, like some unresolv=
ed=20
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=20
analysis,
because the applications to partial differential equations were not=20
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=20
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: catdist@mta.ca de la part de Colin McLarty
Date: ven. 12/09/2008 14:46
=C0: categories@mta.ca
Objet : categories: Re: Bourbaki and Categories
From: zoran skoda
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" (Birkh=E4user 1996) that they did fa=
il.
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" (Dieudonn=E9 "The Work of Nicholas Bourbaki" 1970).
best, Colin
From rrosebru@mta.ca Sun Sep 14 13:59:43 2008 0300
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for categorieslist@mta.ca; Sun, 14 Sep 2008 13:54:36 0300
From: "George Janelidze"
To: Categories
Subject: categories: Noncartesian categorical algebra
Date: Sun, 14 Sep 2008 15:39:11 +0200
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Dear Colleagues,
I would like to make a remark concerning my CT2008 talk.
First let me recall: A lot of mathematics (e.g. of Galois theory) can be
done in the context of adjoint functors between abstract categories with
finite limits  and since one gets all finite limits our of finite products
and equalizers, one can try a further generalization with monoidal structure
plus equalizers. The point was that this seemingly primitive old idea
actually works very seriously and should be taken as the idea of developing
noncartesian categorical algebra. And "noncartesian" is the right idea of
"noncommutative" and "quantum", although what Ross Street means by
"quantum" is more involved and also important. In particular noncartesian
internal categories are to be taken seriously.
At the end of my talk Jeff Egger told us that he knows someone studied such
generalized internal categories, and later sent me an email with the name:
Marcelo Aguiar; and gave the home page address
http://www.math.tamu.edu/~maguiar/ , and... I realized that it is the third
time I am informed about this work! Recently (winter 2007) I spend two very
nice months in Warsaw invited by Piotr Hajac, and discussing mathematics
with him, Tomasz Brzezinski, Tomasz Maszczyk, and a few others  and, among
other interesting things, Tomasz Brzezinski showed me Marcelo Aguiar's
website, including PhD, where those generalized internal categories were
studied. I also recall now an email message from Steven Chase (from 2002)
where he mentions "...the notion of a category internal to a monoidal
category which was developed by my former doctoral student, Marcelo Aguiar,
in his thesis, "Internal Categories and Quantum Groups" (available on
line...".
In fact the whole story begins, in some sense, with the book [S. U. Chase
and M. E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in
Mathematics 97, Springer 1969], which does not use monoidal categories yet,
but very clearly shows that the commutative case is much easier (for Galois
theory) because it makes tensor product (of algebras) (co)cartesian. There
are many other important further contributions by other authors of different
generations. Knowing them personally, I can name Bodo Pareigis, Stefaan
Caenepeel, Peter Schauenburg, and the aforementioned Polish mathematicians
(although Tomasz Brzezinski is in UK now), but I am not ready to give any
reasonably complete list. There are also thingstobecorrected happening:
for instance by far not enough comparisons have been made with the
Australian work on abstract monoidal categories, and some authors use words
like "coring"...
George Janelidze
From rrosebru@mta.ca Mon Sep 15 08:34:23 2008 0300
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Date: Mon, 15 Sep 2008 06:55:55 +0200
From: Andre.Rodin@ens.fr
To: categories@mta.ca
Subject: categories: Re: Bourbaki and Categories
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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 isomor=
phism;
in the structuralist mathematics the notion of isomorphism is basic and t=
he
notion of general morphism is derived (as in Bourbaki). In CT this is th=
e
other way round: the notion of general morphism is basic while isos are d=
efined
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 rea=
sons
is closely related to structuralist mathematics it is not, in my understa=
nding,
a part of structuralist mathematics  at least not if one takes CT *serio=
usly*,
i.e. as foundations.
best,
andrei
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for categorieslist@mta.ca; Mon, 15 Sep 2008 08:28:42 0300
Date: Sun, 14 Sep 2008 13:53:12 0600 (MDT)
Subject: categories: Re: Bourbaki and Categories
From: mjhealy@ece.unm.edu
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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, monad=
s
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 compute=
r
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 u=
p
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 mathematic=
s
teaching that incorporates category theory needs to begin in 6th Grade (i=
n
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
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for categorieslist@mta.ca; Mon, 15 Sep 2008 08:32:12 0300
Date: Mon, 15 Sep 2008 09:58:33 +0200
From: Andree Ehresmann
To: categories@mta.ca
Subject: categories: Re: Bourbaki and Categories
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Dear all,
I would add some information on Bourbaki/categories/France.
Charles Ehresmann has been an active member of Bourbaki from 1936 up =20
to the end of the war, when he began to no more regularly participate =20
and wanted to resign (it was not accepted but replaced by an age limit =20
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 =20
between France and the USA were entirely broken during the war, so =20
that mathematical ideas could not circulate and categories were only =20
heard of after the war, at a time where the more general parts of the =20
treatise were written or at least prepared (the successive versions =20
process was very slow).
Charles said to me that he did not recall to have read Eilenberg =20
& Mac Lane's paper before the fifties, or at least not seen its =20
interest. Naturally he had sooner made a large use of groupoids in is =20
foundation of differential geometry, and he had even defined the =20
general "composition of jets" and given its properties, but without =20
linking it to the notion of a category. He exposed it in a course in =20
Rio de Janeiro in the early fifties, and one of his students =20
(Constantino de Barros who later came to Paris to prepare a thesis =20
with him) suggested that there was a connection with categories. =20
Charles' first large use of categories is in his seminal paper =20
"Gattungen von lokalen Strukturen" (1957, reprinted in "Charles =20
Ehresmann: Oeuvres completes et commentees" Part I).
It is around this date that the word "category" began to circulate =20
in France. In 1957, Choquet (with whom I prepared my thesis) =20
suggested that I learnt more on the notion of category which he did =20
not know but seemed to have many applications (it was the reason for =20
which I first went to see Charles!). It should be noted that Choquet =20
was less conservative than many French mathematicians. In 1959, he =20
defended the development of probabilities by inviting Loomis to give a =20
course (I remember Henri Cartan saying then to PaulAndre Meyer that =20
he should not study this domain for it would be bad for his career!). =20
And later on, he defended Logic which was very badly considered.
A final remark: the "disdain" for categories (not to be confused =20
with 'ignorance') came only later on, since Charles was given the =20
"Prix Petit d'Ormoy" by the French Academy in 1965, essentially for =20
his recent work on categories...
Andree C. Ehresmann
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for categorieslist@mta.ca; Mon, 15 Sep 2008 19:12:52 0300
Date: Mon, 15 Sep 2008 07:59:53 0400 (EDT)
From: Michael Barr
To: Andre.Rodin@ens.fr
cc: categories@mta.ca
Subject: categories: Re: Bourbaki and Categories
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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 (11 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
>
>
>
From rrosebru@mta.ca Mon Sep 15 19:22:04 2008 0300
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for categorieslist@mta.ca; Mon, 15 Sep 2008 19:14:21 0300
From: Joost Vercruysse
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Subject: categories: Re: Noncartesian categorical algebra
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On 14sep08, at 15:39, George Janelidze wrote:
Dear George and all,
> There are also thingstobecorrected happening:
> for instance by far not enough comparisons have been made with the
> Australian work on abstract monoidal categories, and some authors
> use words
> like "coring"...
I hope the following information can be of help here:
Indeed, Marcello Aguilar gave a definition of `internal categories'.
Although the abstract definition of a `coring' looks formally the same
as the one of an internal category (or, if you wish, an internal
cocategory), corings provide examples of these internal cocategories,
but they (usually) refer to a much more concrete situation: a coring
is a comonoid in the monoidal category of bimodules over a given
(possibly noncommutative) ring, this dualizes usual ring extensions.
The theory of corings is in fact quite young, and grew from a pure
algebraic theory to something more and more categorical in the last
few years (this might cause some confusion, `internal corings', which
can be defined in certain monoidal categories (the regular ones from
aguilar) or bicategories, are indeed the same objects as internal
cocategories, there is no need for two names for the same thing at
this level of generality). Therefore, I find the above remark ``not
enough comparision have been made ...'' indeed correct: I believe that
people from corings can learn from more from the pure category theory
side, and hopefully the other way around as well.
Best wishes,
Joost.
From rrosebru@mta.ca Mon Sep 15 19:25:06 2008 0300
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To: categories@mta.ca
Subject: categories: Categories in Algebra, Geometry and Logic  Brussels 10, 11 October 2008
From: Rudger Kieboom
Date: Mon, 15 Sep 2008 18:03:27 +0200
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Dear All,
This is the second announcement (and final announcement
on this categories  mailing list) of the meeting in Brussels
on Friday 10 and Saturday 11 October 2008 in honour of
Francis Borceux and Dominique Bourn on the occasion of
their 60th birthdays.
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D
Categories in Algebra, Geometry and Logic
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D
Friday 10 and Saturday 11 October 2008, at the
Royal Flemish Academy of Belgium for Sciences and the Arts
Paleis der Academie=EBn / Palais des Academies
Hertogsstraat 1 Rue Ducale
1000 Brussel / Bruxelles
Belgium
For all information, and downloadable poster,
see the conference web page:
http://www.math.ua.ac.be/bbdays/
Further (final) announcements (on the
conference dinner, and other practical information)
will be sent by email to the registered participants.
Abstracts of the talks will be made available on the
web page as soon as possible.
Keynote Speakers (55 min. talks):

J. Ad=E1mek (Braunschweig)
J. B=E9nabou (Paris)
M. M. Clementino (Coimbra)
A. Ehresmann (Amiens)
G. Janelidze (Cape Town)
P. T. Johnstone (Cambridge)
F. W. Lawvere (Buffalo)
J. Penon (Paris)
J. Rosick=FD (Brno)
W. Tholen (Toronto)
Shorter Communications (25 min. talks):

S. Caenepeel (Brussel)
Z. Janelidze (Cape Town)
S. Mantovani (Milano)
D. Rodelo (Faro)
I. Stubbe (Antwerpen)
T. Van der Linden (Coimbra, Brussel)
From rrosebru@mta.ca Mon Sep 15 19:26:52 2008 0300
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Date: Mon, 15 Sep 2008 20:26:56 +0100 (BST)
From: Dusko Pavlovic
To: categories@mta.ca
Subject: categories: Re: Bourbaki and Categories
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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 wikimethods 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
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Subject: categories: Re: Bourbaki and Categories
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I agree with Andre. Encapsulating a group of mathematicians inside a =20=
single named entity fosters a kind of collaborative spirit in which =20
good ideas are not kept for personal use later but are shared amongst =20=
the community. When ideas are shared in real time, good mathematics =20
can be produced faster. Anyone who wants to join the collective can =20
do so, and the collective produces highly useful material. Of course =20=
such an enterprise is orthogonal to namerecognition, and maybe to =20
getting tenure! But there is certainly something good about it, as =20
there is about wikipedia and the open source movement.
I also agree that the internet could be used in a better way to =20
transfer knowledge of mathematics. Math papers are written linearly, =20=
in the bottomup (Euclid/Bourbaki) style, to some extent. Whereas =20
words on paper are in this sense onedimensional, computers offer =20
many more dimensions for knowledge transfer.
Even more interesting to me would be a kind of zoomfeature on =20
proofs. Proofs are in the eye of the beholder: for example it has =20
been debated as to whether Perelman's 70 pages was a full proof of =20
geometrization. Given a proof with a statement which one does not =20
understand, a mathematician may find himself reproving something that =20=
was obvious to (or wrongly assumed to be obvious by) another =20
mathematician. The community could benefit if a mathematician who =20
proves such a statement then uploaded the proof, even in rough form, =20
to some kind of math wiki. If it were wellorganized, this math wiki =20=
could revolutionize how mathematics is done. In fact, choosing the =20
"right way" to organize such a site may itself be a problem which =20
could produce interesting mathematics.
Whatever the case may be, I am all for the idea of a new Bourbaki=20
style enterprise in some form or another. I think it may first =20
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 =20
> important.
> I recall asking the question to Samuel Eilenberg 25 years ago and =20
> more recently to Pierre Cartier.
> If my recollection is right, Bourbaki had essentially two options: =20
> 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 =20
> task of rewriting everything.
> Other factors may have contributed on a smaller scale, like some =20
> 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 =20=
> 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 =20
> functional analysis,
> because the applications to partial differential equations were not =20=
> 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 =20
> research.
>
> Bourbaki is dead but I hope that the humanistic philosophy behind =20
> 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 =20
> 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
>
>
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for categorieslist@mta.ca; Mon, 15 Sep 2008 19:21:21 0300
Date: Tue, 16 Sep 2008 06:58:46 +1000
Subject: categories: Re: Another terminological question...
From: Steve Lack
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Dear Jeff,
I had a chat about this with a couple of other longtime users of the
terms tensor and cotensor (Ross Street and Dominic Verity). We
all think that, given the current overburdening of the word tensor,
this would be a sensible change.
Regards,
Steve Lack.
On 12/09/08 7:56 PM, "Jeff Egger" wrote:
> Dear all,
>
> In ``basic concepts of enriched category theory'',
> Kelly writes:
>
>> Since the conetype limits have no special position of
>> dominancein the general case, we go so far as to call
>> weighted limits simply ``limits'', where confusion
>> seems unlikely.
>
> My question is this: why does he not apply the same
> principle to the concept of powers? Instead, he
> introduces the word ``cotensor'', apparently in order
> to reserve the word ``power'' for that special case
> which could sensibly be called ``discrete power''.
> [This leads to the unfortunate scenario that a
> ``cotensor'' is a sort of limit, while dually a
> ``tensor'' is a sort of colimit.] Is there perhaps
> some genuinely mathematical objection to calling
> cotensors powers (and tensors copowers) which I may
> have overlooked?
>
> Cheers,
> Jeff.
>
> P.S. I specify ``genuinely mathematical'' because I
> know that some people are opposed to any change of
> terminology for any reason whatsoever. Obviously,
> I disagree; in particular, I don't see that minor
> terminological schisms such as monad/triple (even
> compact/rigid/autonomous) are in any way detrimental
> to the subject.
>
> I also disagree with the notion (symptomatic of the
> curiously feudal mentality which seems to permeate the
> mathematical community) that prestigious mathematicians
> have more right to set terminology than the rest of us.
> I see no correlation between mathematical talent and
> good terminology; nor do I understand that a great
> mathematician can be ``dishonoured'' by anything less
> than strict adherence to their terminologyor notation,
> for that matter.
>
>
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for categorieslist@mta.ca; Tue, 16 Sep 2008 21:05:52 0300
From: "George Janelidze"
To:
Subject: categories: Re: Bourbaki and Categories
Date: Tue, 16 Sep 2008 02:03:12 +0200
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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 categorytheorists 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
From rrosebru@mta.ca Tue Sep 16 21:13:48 2008 0300
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for categorieslist@mta.ca; Tue, 16 Sep 2008 21:07:07 0300
Date: Tue, 16 Sep 2008 08:52:36 +0200
From: Andrej Bauer
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Subject: categories: Re: Bourbaki and Categories
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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
"hivescience", 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
From rrosebru@mta.ca Tue Sep 16 21:16:23 2008 0300
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for categorieslist@mta.ca; Tue, 16 Sep 2008 21:10:07 0300
Date: Tue, 16 Sep 2008 01:57:50 0700
From: Vaughan Pratt
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Subject: categories: Re: Bourbaki and Categories
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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 reeducation
before it becomes involuntary.)
Vaughan Pratt
From rrosebru@mta.ca Tue Sep 16 21:18:10 2008 0300
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for categorieslist@mta.ca; Tue, 16 Sep 2008 21:12:06 0300
Date: Tue, 16 Sep 2008 12:27:07 +0200
From: Andre.Rodin@ens.fr
To: categories@mta.ca
Subject: categories: Re: Bourbaki and Categories
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When one defines, say, a group =E0 la Borbaki, i.e. structurally, it usua=
lly goes
without saying that the defined structure is defined up to isomorphism. T=
he
notion of isomorphism plays in this case the role similar to that of equa=
lity
in the (naive) arithmetic. In most structural constexts the distinction b=
etween
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 no=
tion
of admissible map, say, that of group homomorphism, on the contrary, requ=
ires a
definition, which may be nontrivial.
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 th=
is
sense the modern axiomatic method is structuralist. In his oftenquoted l=
etter
to Frege Hilbert explicitely says that a theory is "merely a framework" w=
hile
domains of their objects are multiple and transform into each other by
"univocal and reversible oneone transformations". Those who trace the hi=
story
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 i=
s
Functorial Semantics, which brings a *category* of models, not just one m=
odel
up to isomorphism. From the structuralist viewpoint the presence of
nonisomorphic models (i.e. noncategoricity) is a shortcoming of a given
theory. From the perspective of Functorial Semantics it is a "natural" fe=
ature
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 Hilbe=
rt's
basic structuralist intuition applies in this case. In my understanding t=
hings
work in Sketch theory more like in Euclid. Think about circle and straigh=
t 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" fro=
m a new
categorical viewpoint. Bourbaki took Euclid's "Elements" as a model for h=
is
work just like did Hilbert writing his "Gundlagen". In my view, this is t=
his
longterm 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 Hilb=
ert's
"Grundlagen". It covers very little  actually near to nothing  of geome=
try of
its time. But at the same time it provided a very powerful model of how t=
o do
mathematics in a new way, which greatly influenced mathematics education =
and
mathematical research in 20th century. In my view, Euclid's "Elements" an=
d
Hilbert's "Grundlagen" are better examples to be followed.
best,
andrei
le 15/09/08 12:59, Michael Barr =E0 barr@math.mcgill.ca a =E9crit :
> 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 Harriso=
n
> arrived in 1959) and each of them started by defining an appropriate
> notion of "admissible map". I do not recall any special point being ma=
de
> of isomorphism and I think in general it was used for what we now call =
a
> bimorphism (11 and onto) even in cases, such as topological groups, wh=
en
> they were not isomorphisms.
>
> To be sure Bourbaki was not mentioned either, but this structuralist
> influence seemed strong.
>
> Michael
>
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