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Subject: categories: quotients of groupoids
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This is a comment on one aspect of Jean Pradines' very interesting
posting of June 8. I quote a large part of it for reference:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
EXTRACT FROM JEAN'S POSTING:
Now what I really had in mind was not the purely algebraic (rather obviou=
s)
setting, but the study of quotients for the differentiable groupoids
(nowadays called smooth or Lie) introduced by Ehresmann in 1959, which ar=
e
groupoids in the category of manifolds.
The statement I had obtained (which again certainly cannot by any means b=
e
deduced from the general results of Ehresmann concerning topological or
structured groupoids) was published in 1966 in my Note (prealably submitt=
ed to Ehresmann and transmitted by him) :
-Th=E9orie de Lie pour les groupo=EFdes diff=E9rentiables. Relations entr=
e
propri=E9t=E9s locales et globales, CRAS (Paris), s=E9rie A, t.263, p.907=
-910, 19
d=E9cembre 1966-
in Th=E9or=E8me 5 (in this statement the Bourbaki term "subimmersion" is
improperly used and has to be understood in the more restricted acception
"submersion onto a submanifold" ; note also that the implication from 2=B0=
to 1=B0 is valid only
when the fibres of the domain map of H are connected). (Note that in this=
statement the algebraic theory of the two-sided quotient of a groupoid b=
y an invariant subgroupoid is implicitely considered as "well known" with=
out reference. This was in fact a diplomatic consequence of the above-men=
tionned conversation ! At least at that period it seems absolutely certai=
n that no reference did exist, and probably very few people, if any, had =
had the opportunity of thinking to that sort of questions, since the main=
stream of categoricists were despising groupoids as beig trivial, since =
equvalent to groups).
The unpublished proof of this statement relied on a careful and rather
delicate study of the foliation defined by the two-sided cosets.
The very elegant proof for the case of Lie groups given in Serre, Lie
Algebras and Lie Groups (lecture in Harvard, 1964, p.LG 4.10-11), relying=
on the so-called Godement's
theorem, works only for one-sided cosets and yields only the too special =
case
above-mentioned or more generally the statement of Proposition 2 in the
previous Note (which extends to groupoids the classical theory of homogen=
eous spaces for possibly non invariant subgroups).
It is clear that this last proof may be immediately written in a purely
diagrammatic way and remains valid in much more general contexts when an
abstract Godement's theorem is available. It turns out that this is the c=
ase
for most of (perhaps almost all) the categories considered by "working
mathematicians", notably the abelian categories as well as toposes, the
category of topological spaces (with a huge lot of variants), the
category of Banach spaces, and many useful categories that are far from b=
eing complete.The precise definitions for what is meant by an abstract Go=
dement's theorem were given in my paper :
-Building Categories in which a Godement's theorem is available-
published in the acts of the Second Colloque sur l'Alg=E8bre des Cat=E9go=
ries,
Amiens 1975, Cahiers de Topologie....(CTGDC).
In this last paper I introduced the term "dyptique de Godement" for a
category in which one is given two subcategories of "good mono's" and "go=
od
epi's" (playing the roles of embeddings and surmersions in Dif) such that=
a formal Godement's theorem is valid.
These considerations explain why I was strongly motivated for adapting
Serre's diagrammatic proof to the case of two-sided cosets (since such a =
proof would immediately extend the theory of quotients for groupoids in v=
arious non complete categories used by "working mathematicians", yielding=
a lot of theorems completely out of the range of Ehresmann's general the=
ory of structured groupoids). However this was
achieved only in 1986 in my Note :
-Quotients de groupo=EFdes diff=E9rentiables, CRAS (Paris), t.303, S=E9ri=
e I, 1986, p.817-820.-
The proof requires the use of certain "good cartesian squares" or "good p=
ull back
squares", which, though they are not the most general pull back's existin=
g
in the category Dif of manifolds, cannot be obtained by the (too
restrictive) classical condition of transversality. Though written in the
framework of the category Dif (in order not to frighten geometers, but wi=
th
the risk of frightening categoricists), this paper is clearly thought in
order to be easily generalizable in any category where a suitable set of
distinguished pull back's is available , assuming only some mild stabilit=
y properties.
In this paper I introduced the seemingly natural term of "extensors" for
naming those functors between (structured) groupoids which arise from the
canonical projection of a groupoid onto its quotient by a normal
subgroupoid. Equivalent characterizations are given.(I am ignoring if ano=
ther term is being used in the literature).This notion is resumed and use=
d in my paper :
-Morphisms between spaces of leaves viewed as fractions-
CTGDC, vol.XXX-3 (1989),p. 229-246
which again is written in the smooth context, but using purely diagrammat=
ic descriptions (notably for Morita equivalences and generalized morphism=
s) allowing immediate extensions for various categories.
As a prolongation of this last paper, I intend in future papers to give a
description of the category of fractions obtained by inverting those
extensors with connected fibres. This gives a weakened form of Morita
equivalence which seems basic for understanding the transverse structure =
of
foliations with singularities.
<<<<<<<<<<<<<<<<<<<<
In response to Jean's paper
[QGD] Quotients de groupo=EFdes diff=E9rentiables, CRAS (Paris), t.303,=20
S=E9rie I, 1986, p.817-820.=20
Philip Higgins and I wrote two papers ([HM90a] and [HM90b] below)=20
dealing with general quotients of Lie (=3Ddifferentiable) groupoids=20
and Lie algebroids. Our starting point was the idea that:
``a morphism in a given category is entitled to be called a quotient=20
map if and only if it is entirely determined by data on the domain''
(Perhaps this will seem naive to true categorists, but I write as an=20
end user of category theory, not a developer.)=20
I would summarize [QGD] as proving that the `regular extensors' are
quotient maps in the category of Lie groupoids. [HM90b] then showed,
by extending the notion of kernel, that in fact all extensors are
quotient maps in the category of Lie groupoids.
Philip and I used the term `fibration' for what Jean calls an
`extensor'; these maps satisfy a natural smooth version of the=20
notion of `fibration of groupoids' introduced by Ronnie Brown=20
in 1970 (building on work of Frolich, I think). (Throughout this=20
post, I assume base maps to be surjective submersions.)
There is unlikely to be a more general class of quotient maps for
Lie groupoids: the fibration condition on a groupoid morphism=20
$F : G \to H$ is exactly what is needed to ensure that any product=20
in the codomain is determined by a product in the domain: given=20
elements $h, h'$ of $H$ which are composable, one wants to be able=20
to write $h =3D F(g)$ and $h' =3D F(g')$ in such as way that $gg'$ will=20
exist and determine $hh'$; the fibration condition is the weakest=20
simple condition which ensures this.=20
Fibrations of Lie groupoids are not determined by their kernel in=20
the usual sense (=3D union of the kernels of the maps of vertex=20
groups); one also requires the kernel pair of the base map, and the=20
action of this (considered as a Lie groupoid) on the manifold of=20
one--sided cosets of the domain. This data, which Philip and I=20
called a `kernel system' is equivalent to a suitably well--behaved
congruence on the domain groupoid. Though more complicated than the
usual notion of kernel, the notion of normal subgroupoid system=20
gives an exact extension of the `First Isomorphism Theorem'.=20
We also showed that regular fibrations (=3D Jean's regular extensors)
are precisely those in which the additional data can be deduced from
the standard kernel and the base map; in the regular case the two=20
step quotient can be reduced to a single quotient consisting of=20
double cosets (as Jean remarks in his post). The congruences=20
corresponding to regular fibrations are those which, regarded as=20
double groupoids, satisfy a double source condition.=20
A good example of a fibration which is not regular is the division=20
map $\delta: (g,h)\mapsto gh^{-1}$ in a group $G$, considered as a=20
groupoid morphism from the pair groupoid $G\times G$ to the group $G$.
All of this (for Lie algebroids, as well as for Lie groupoids and=20
for vector bundles) is in the two papers referenced below. There is=20
also a full account coming in my book `General Theory of Lie=20
groupoids and Lie algebroids' (CUP) which should be appearing=20
in the next few months.=20
@ARTICLE{HM90a,
author =3D {P.~J. Higgins and K.~C.~H. Mackenzie},
title =3D {Algebraic constructions in the category of {L}ie=20
algebroids},
journal =3D {J.~Algebra},
year =3D 1990,
volume =3D 129,
pages =3D "194-230",
}
@ARTICLE{HM90b,
author =3D {P.~J. Higgins and K.~C.~H. Mackenzie},
title =3D {Fibrations and quotients of differentiable groupoids},
journal =3D {J.~London Math. Soc.~{\rm (2)}},=20
year =3D 1990,
volume =3D 42,
pages =3D "101-110",
}
Kirill Mackenzie
http://www.shef.ac.uk/~pm1kchm/
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Subject: categories: [Csl03] Call for Participation
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We apologize for multiple copies of this call for participation
Annual Conference of the European Association for
Computer Science Logic
and
8th Kurt Goedel Colloquium
CSL'03 & KGC
25 August afternoon - 30 August 2003, Vienna, Austria
http://www.logic.at/csl03
Computer Science Logic (CSL) is the annual conference of the European
Association for Computer Science Logic (EACSL). The Kurt Goedel
Colloquium (KGC) is the biennial conference of the Kurt Goedel
Society (KGS). The joint conference is intended for computer scientists
whose research activities involve logic, as well as for logicians
working on issues significant for computer science.
LIST OF ACCEPTED PAPERS
A Fixed-Point Logic with Symmetric Choice
(Anuj Dawar and David Richerby)
A Logic for Probability in Quantum Systems
(Ron van der Meyden, Manas Patra)
A Strongly Normalising Curry-Howard correspondence for IZF Set Theory
(Alexandre Miquel)
Atomic Cut Elimination for Classical Logic
(Kai Br=FCnnler)
Automata on Lempel-Ziv-compressed strings
(Hans Leiss, Michel de Rougemont)
Bistability: an extensional characterization of sequentiality.
(James Laird)
Calculi of Meta-variables
(Masahiko Sato, Takafumi Sakurai, Yukiyoshi Kameyama, Atsushi Igarashi)
Comparing the succinctness of monadic query languages over finite trees
(Martin Grohe, Nicole Schweikardt)
Complexity of Some Problems in Modal and Intuitionistic Calculi
(Larisa Maksimova and Andrei Voronkov)
Computational Aspects of $\Sigma$-definability over the Real Numbers
without the Equality Test
(Margarita Korovina)
Concurrent Construction of Proof-Nets
(Jean-Marc Andreoli, Laurent Mazar=E9)
Constraint Satisfaction with Countable Homogeneous Templates
(Manuel Bodirsky and Jaroslav Nesetril)
Coping polynomially with numerous but identical elements within planning
problems.
(Max Kanovich and Jacqueline Vauzeilles)
Deciding Monotonic Games
(Parosh Aziz Abdulla, Ahmed Bouajjani, Julien d'Orso)
Extending the Dolev-Yao Intruder for Analyzing an Unbounded Number of
Sessions
(Yannick Chevalier, Ralf Kusters, Michael Rusinowitch,
Mathieu Turuani, Laurent Vigneron)
Generating all Abductive Explanations for Queries on Propositional Horn
Theories
(Thomas Eiter, Kazuhisa Makino)
Goal-Directed Calculi for Goedel-Dummett Logics
(George Metcalfe, Nicola Olivetti and Dov Gabbay)
Henkin models of the partial lambda-calculus
(Lutz Schr=F6der)
Logical relation for dynamic name creation
(Yu ZHANG, David NOWAK)
Machine Characterizations of the Classes of the W-hierarchy.
(Yijia Chen and Joerg Flum)
Modular Semantics and Logics of Classes
(Bernhard Reus)
More computation power for a denotational semantics for first order logic
(C.F.M. Vermeulen)
Nominal Unification
(Christian Urban, Andrew Pitts, Murdoch Gabbay)
On Algebraic Specifications of Abstract Data Types
(Bakhadyr Khoussainov)
On Complexity gaps for Resolution-based proof systems
(Stefan Dantchev and S=F8ren Riis)
On the Complexity of Existential Pebble Games
(Phokion G. Kolaitis and Jonathan Panttaja)
Parity of Imperfection
(J. C. Bradfield)
Pebble games on trees
(Lukasz Krzeszczakowski)
Positive Games and Persistent Strategies
(Jacques Duparc)
Program Complexity of Dynamic LTL Model Checking
(Detlef K=E4hler, Thomas Wilke)
Quantified Constraints: Algorithms and Complexity
(Ferdinand Boerner, Andrei Bulatov, Peter Jeavons, Andrei Krokhin)
Refined Complexity Analysis of Cut Elimination
(Philipp Gerhardy)
Simple stochastic parity games
(Krishnendu Chatterjee, Thomas A. Henzinger, and Marcin Jurdzinski)
Strong normalization of the typed $\lambda_{ws}$- calculus
(Ren=E9 DAVID & Bruno Guillaume)
The Arithmetical Complexity of Dimension and Randomness
(John M. Hitchcock, Jack H. Lutz, Sebastiaan A. Terwijn)
The Commuting V-Diagram: On the Relation of Refinement and Testing
(Bernhard K. Aichernig)
The Surprising Power of Restricted Programs and G=F6del's Functionals
(Lars Kristiansen, Paul J. Voda)
Towards a proof system for admissibility
(R. Iemhoff)
Validity of CTL Queries Revisited
(Marko Samer, Helmut Veith)
Registration is through the Internet:
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The registration fee (400 euro) includes the costs for the conference
dinner, the reception, and a copy of the conference proceedings. The
registration fee does not include the costs for the excursion to Wachau.
INVITED SPEAKERS:
Sergei Artemov (CUNY, USA) [jointly invited by CSL'03 & KGC and
ESSLLI (European Summer School in Logic Language and Information)]
Bruno Buchberger (Johannes Kepler University, Austria)
Dov Gabbay (King's College London, England)
Helmut Veith (Vienna University of Technology, Austria)
Nikolai Vorobjov (University of Bath, England)
Andrei Voronkov (University of Manchester, England)
TUTORIALS:
Verification of infinite state systems
(Ahmed Bouajjani, University of Paris 7, France)
Computational epsilon calculus
(Georg Moser, University of Muenster, Germany, and
Richard Zach, University of Calgary, Canada)
Quantifier elimination
(Nikolai Vorobjov, University of Bath, England)
Winning strategies and controller synthesis
(Igor Walukiewicz, University of Bordeaux, France)
INTERNATIONAL PROGRAM COMMITTEE
Matthias Baaz (chair), Vienna University of Technology
Arnold Beckmann, Vienna University of Technology
Lev Beklemishev, Steklov Inst., Moscow / Utrecht Univ.
Agata Ciabattoni, Vienna University of Technology
Kousha Etessami, University of Edinburgh, Informatics
Chris Fermueller, Vienna University of Technology
Didier Galmiche, LORIA - UHP Nancy
Harald Ganzinger, Max Plank Institut (Saarbr=FCcken)
Erich Gr=E4del, Aachen University
Petr Hajek, Institute for Computer Science, Chech Academy of Science (Prag=
ue)
Martin Hyland, DPMMS , University of Cambridge
Reinhard Kahle, DI, Universidade Nova de Lisboa
Helene Kirchner, LORIA CNRS
Daniel Leivant, Indiana University Bloomington
Johann Makowsky (co-chair), Technion-IIT, Haifa
Jerzy Marcinkowski, Wroclaw University
Franco Montagna, Department of Mathematics, Siena
Robert Nieuwenhuis, Tech. Univ. Catalonia (Barcelona)
Michel Parigot, CNRS / Universit=E9 Paris 7
Jeff Paris, Manchester University, Mathematics Dept
Helmut Schwichtenberg, Munich University, Dept. of Mathematics
Jerzy Tiuryn, Warsaw University
LOCAL ORGANIZING COMMITTEE
Matthias Baaz, Chair
Arnold Beckmann
Agata Ciabattoni
Christian Ferm=FCller
Rosalie Iemhoff
Norbert Preining
Sebastiaan Terwijn
COLOCATED EVENTS
CoLogNET (23 August)
Workshop on Logic-based Methods for Information Integration
See http://www.kr.tuwien.ac.at/colognet_ws/
GAMES (30 August - 2 September)
2nd Annual Workshop of the European Research Training Network
GAMES (Games and Automata for Synthesis and Validation).
See http://www.games.rwth-aachen.de.
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Date: Fri, 4 Jul 2003 19:05:53 +0200 (CEST)
Subject: categories: Compatibility of functors with limits
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I have the sensation that I'm about to ask a question to which half the
readers of this list will be able to see an answer immediately.
Unfortunately, I'm one of the other half.
What should it mean for a functor to "respect limits"? Consider the
following informal definition: a functor respects limits if given any
diagram in the domain category, the limit of the image of the diagram is no
bigger than it needs to be. Formally, let F: A ---> B be a functor, where
B is a category with (for sake of argument) all small limits and colimits.
Let I be a small category and D: I ---> A a diagram in A; write Cone(D) for
the category of cones on D in A, write Cone(FD) for the category of cones
on FD in B, and write
F_*: Cone(D) ---> Cone(FD)
for the induced functor. Then F can be said to "respect limits for D" if
the colimit of F_* is the terminal object of Cone(FD) (that is, the limit
cone on FD).
* Example: if D has a limit in A then the limit is a terminal object of
Cone(D), so F respects limits for D if and only if it preserves the limit
in the usual sense.
* Example: let B = Set and let A be the category consisting of a pair of
parallel arrows; a functor F: A ---> B consists of sets and functions
sigma, tau: F_0 ---> F_1.
The condition that F respects pullbacks says that sigma and tau are monic
and that the images of sigma and tau are disjoint.
The thought behind "no bigger than it needs to be" (a very approximate
description, I know) is that if we have a cone on D with vertex v then
there's an induced map from F(v) to lim(FD), which in some sense places a
"lower bound" on lim(FD): e.g. if B = Set and F(v) is nonempty then lim(FD)
is nonempty. For F to respect limits for D means that lim(FD) is built up
freely from these F(v)s.
So the question is: is this notion of "respecting limits" well-known or
well-understood? Is there, for instance, some way of rephrasing it that
brings it into more familiar territory?
Thanks very much,
Tom
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Message-ID: <002101c342e1$8a9c9be0$edd8f8c1@wanadoo.fr>
From: "jpradines" <jpradines@wanadoo.fr>
To: <categories@mta.ca>
References: <Pine.LNX.4.33.0307011032190.16268-100000@makar.shef.ac.uk>
Subject: categories: Re: quotients of groupoids SECUND PART
Date: Sat, 5 Jul 2003 12:37:41 +0200
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This is is the announced secund part added to my response to Kirill's
comments concerning terminology.
The first part deals with the relation between my "regular extensors" and
Kirill's "regular fibrations", and points a misunderstanding.
This secund part cannot be read independently of the first one and will
concern Kirill's use of the term "quotient of groupoids".
>From his message quoted below, I now extract the following :
<<``a morphism in a given category is entitled to be called a quotient
map if and only if it is entirely determined by data on the domain''>>
Perhaps this will seem naive to true categorists, but I write as an
end user of category theory, not a developer.) >>
<<There is unlikely to be a more general class of quotient maps for
Lie groupoids: the fibration condition on a groupoid morphism
$F : G \to H$ is exactly what is needed to ensure that any product
in the codomain is determined by a product in the domain: given
elements $h, h'$ of $H$ which are composable, one wants to be able
to write $h =3D F(g)$ and $h' =3D F(g')$ in such as way that $gg'$ will
exist and determine $hh'$; the fibration condition is the weakest
simple condition which ensures this. >>
First I observe that the expression "entirely determined by data on the
domain'' can hardly be given a precise mathematical or logical status, sinc=
e
it seems difficult to justify in a formalisable way why the (tautological)
data of the given
functor f (from H to G) itself could not be considered as data "on H", whil=
e
the extra data of an equivalence relation, a subgroupoid, or a "kernel
system",
would be such.
On the other hand a morphism such as,for instance, the diagonal map H --> H
x H is
certainly entirely determined by the single data H without any extra data !=
!
Precise definitions of the various possible notions of quotient categories
and a thorough discussion can be found in Ehresmann's book, at least in
theory.
The case of groupoids is treated as a special case, but, as pointed in my
first message to Marco, Ehresmann always missed the very simple and
important (though or because) special case of what I call extensors ( =3D
regular fibrations).
This study is made for categories and also for a weaker notion called
"multiplicative
graphs" (in which the associativity is dropped and the composite of two
adjacent arrows may be undefined). The stubborn Ehresmann's reader might
perhaps guess that this notion was probably introduced precisely in order t=
o
handle the composition laws one gets when studying equivalence relations on
a category which are compatible in a natural sense with the algebraic data.
When such a quotient law defines a true category law (which is not
automatic), this one is called a "strict quotient". There exist quotients
which are not strict, but satisfy
the natural universal property of quotients.The condition of what I call
"exactors"
(=3D fibrations) is given as sufficient for a quotient being strict.
However it is very difficult to extract these (interesting and important)
facts from the book because of the very clumsy notations, terminology and
redaction, and the (intentional !) absence of examples.
So, instead of asking the reader to try to read Ehresmann's book, it will
be enough for a clear understanding to give some very simple examples of
what may happen, being content with the case of groupoids, in order to
illustrate Ehresmann's definitions and their motivations (I confess that I
have no idea of the examples Ehresmann had in mind, and I don't claim I
understood everything ! ).
First we give an alternative definition for strict quotients as resulting
from criteria proved in Ehresmann's book :
a quotient is strict when the map Vf : VH --> VG (where the functorial
symbol V is used
here, unusually, for denoting composable pairs or pairs with the same domai=
n
or the same codomain) is surjective (this allows an immediate transfer to
the smooth case via the diptych policy).
In the following examples, we'll denote by I (resp. D (written for Delta,
pictured by a triangle)) the banal (=3D coarse) groupoid with just two (res=
p.
three) objects.
FIRST EXAMPLE :
Let H be the groupoid III (sum of three copies of I), G =3D D, and f be the
map sending the three copies of I respectively onto the three edges of the
triangleD. (Note that all these groupoids may be considered as topological
or even smooth withe the discrete topology, and hence f as continuous or
smooth).Then :
-1) f defines a surjective functor ; let R be the equivalence relation on H
thus defined.
-2) f is faithful, but not full, and not an exactor.
-3) R is "bicompatible" in Ehresmann's sense, wich means compatible with th=
e
composition law of H as well as with the source and range (otherwise domain
and codomain) maps.
-4) The quotient law thus defined on G is only a part of the groupoid law
(it
defines a "multiplicative graph" in the sense of Ehresmann).
So H is not a "strict quotient" in Ehresmann's sense.
-5) However this groupoid law is "entirely determined" by the quotient law
(hence by H and R) in that sense that it is the unique groupoid law
extending that quotient law and defining a quotient in the universal sense
(more precise statements can be found in Ehresmann's book).
So f should be called a quotient map by Kirill according to the first
quotation (one does not see why the data of the equivalence relation R,whic=
h
determines eveything, could not be considered as data "on H"). But this
contradicts the secund one.
-6) (f,G) is a quotient groupoid of H in the very general and widely
admitted (not
only by Ehresmann, but by Bourbaki and nearly everybody) sense of quotient
structures, which means here that it satisfies the following universal
property of quotients :
given any groupoid Z and any functor h from H to Z admitting a set
theoretical factorization h =3D gf, then g defines a functor from G to Z.
[Exercise : show that f is composed of an (injective) groupoid equivalence,
and a surjective actor. Hint : embed suitably H =3D III into K =3D DDD (sum=
of
three
copies of D), and note that K may be viewed as an action groupoid.]
SECUND EXAMPLE :
We now define H by suppressing a copy of I in the previous example and
restricting f.
Nothing is changed in the conclusions save the surjectivity of f, so that
now G is no more set-theoretic quotient, but is still a quotient in the
categorical sense.
THIRD EXAMPLE :
We start again with III, but define H by now adding a copy of D and
extending f obviously (i.e. by the identity).
Then what is changed in the conclusions is that now the groupoid law of G
may be fully defined as the quotient law (since composable arrows of G are
now always images of some composable arrows of H), so that (f,G) is a
"strict
quotient" of H in the sense of Ehresmann, though f is not an exactor. Once
again
it should be a quotient in the sense of Kirill according to the first
(informal) "definition", though it still contradicts the secund one.
So we see that we have a chain of strict implications :
quotient (in the universal sense) <=3D=3D surjective quotient <=3D=3D stric=
t
quotient <=3D=3D surjective exactors (alias fibrations) <=3D=3D extensors (=
alias
regular fibrations).
I think there is absolutely no reason to keep the word "quotient map", for
the fourth term of this chain, which is just a criterion for quotients amon=
g
others. Moreover among the exactors (star surjectivity condition alone),
surjective exactors (where the surjectivity is a consequence of the
surjectivity for the bases) are just a special case : the importance of
exactors comes mainly, I think, from the fact that any functor f is
isomorphic with an exactor (which is surjective iff f is essentially
surmersive) (see Prop. 8.1, 8.2 and 10.4 of [MVF]) and all of this extends
to the smooth , via the general "diptych" policy.
JEAN PRADINES
----- Original Message -----
From: <K.Mackenzie@sheffield.ac.uk>
To: Categories List <categories@mta.ca>
Sent: Tuesday, July 01, 2003 11:33 AM
Subject: categories: quotients of groupoids
This is a comment on one aspect of Jean Pradines' very interesting
posting of June 8.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
=2E........................................................................=
=2E..
=2E...........
<<<<<<<<<<<<<<<<<<
In response to Jean's paper
[QGD] Quotients de groupo=EFdes diff=E9rentiables, CRAS (Paris), t.303,
S=E9rie I, 1986, p.817-820.
Philip Higgins and I wrote two papers ([HM90a] and [HM90b] below)
dealing with general quotients of Lie (=3Ddifferentiable) groupoids
and Lie algebroids. Our starting point was the idea that:
``a morphism in a given category is entitled to be called a quotient
map if and only if it is entirely determined by data on the domain''
(Perhaps this will seem naive to true categorists, but I write as an
end user of category theory, not a developer.)
I would summarize [QGD] as proving that the `regular extensors' are
quotient maps in the category of Lie groupoids. [HM90b] then showed,
by extending the notion of kernel, that in fact all extensors are
quotient maps in the category of Lie groupoids.
Philip and I used the term `fibration' for what Jean calls an
`extensor'; these maps satisfy a natural smooth version of the
notion of `fibration of groupoids' introduced by Ronnie Brown
in 1970 (building on work of Frolich, I think). (Throughout this
post, I assume base maps to be surjective submersions.)
There is unlikely to be a more general class of quotient maps for
Lie groupoids: the fibration condition on a groupoid morphism
$F : G \to H$ is exactly what is needed to ensure that any product
in the codomain is determined by a product in the domain: given
elements $h, h'$ of $H$ which are composable, one wants to be able
to write $h =3D F(g)$ and $h' =3D F(g')$ in such as way that $gg'$ will
exist and determine $hh'$; the fibration condition is the weakest
simple condition which ensures this.
Fibrations of Lie groupoids are not determined by their kernel in
the usual sense (=3D union of the kernels of the maps of vertex
groups); one also requires the kernel pair of the base map, and the
action of this (considered as a Lie groupoid) on the manifold of
one--sided cosets of the domain. This data, which Philip and I
called a `kernel system' is equivalent to a suitably well--behaved
congruence on the domain groupoid. Though more complicated than the
usual notion of kernel, the notion of normal subgroupoid system
gives an exact extension of the `First Isomorphism Theorem'.
We also showed that regular fibrations (=3D Jean's regular extensors)
are precisely those in which the additional data can be deduced from
the standard kernel and the base map; in the regular case the two
step quotient can be reduced to a single quotient consisting of
double cosets (as Jean remarks in his post). The congruences
corresponding to regular fibrations are those which, regarded as
double groupoids, satisfy a double source condition.
A good example of a fibration which is not regular is the division
map $\delta: (g,h)\mapsto gh^{-1}$ in a group $G$, considered as a
groupoid morphism from the pair groupoid $G\times G$ to the group $G$.
All of this (for Lie algebroids, as well as for Lie groupoids and
for vector bundles) is in the two papers referenced below. There is
also a full account coming in my book `General Theory of Lie
groupoids and Lie algebroids' (CUP) which should be appearing
in the next few months.
@ARTICLE{HM90a,
author =3D {P.~J. Higgins and K.~C.~H. Mackenzie},
title =3D {Algebraic constructions in the category of {L}ie
algebroids},
journal =3D {J.~Algebra},
year =3D 1990,
volume =3D 129,
pages =3D "194-230",
}
@ARTICLE{HM90b,
author =3D {P.~J. Higgins and K.~C.~H. Mackenzie},
title =3D {Fibrations and quotients of differentiable groupoids},
journal =3D {J.~London Math. Soc.~{\rm (2)}},
year =3D 1990,
volume =3D 42,
pages =3D "101-110",
}
Kirill Mackenzie
http://www.shef.ac.uk/~pm1kchm/
From rrosebru@mta.ca Sun Jul 6 13:22:47 2003 -0300
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Reply-To: "jpradines" <jpradines@wanadoo.fr>
From: "jpradines" <jpradines@wanadoo.fr>
To: <categories@mta.ca>
References: <Pine.LNX.4.33.0307011032190.16268-100000@makar.shef.ac.uk>
Subject: categories: Re: quotients of groupoids (K. Mackenzie's comments on J. Pradines'answer to Marco Mackaay)
Date: Sat, 5 Jul 2003 00:55:02 +0200
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This is a purely terminological remark concerning the comments sent by =
Kirill Mackenzie about the response I made to Marco Mackaay's message =
("reference normal categorical subgroup ?", June 5). From these =
comments, fully quoted below (though omitting the partial quotations =
from my own message of June 8), I extract the following two fragments :
<<Philip and I used the term `fibration' for what Jean calls =
an`extensor'>>
<<regular fibrations (=3D Jean's regular extensors) are precisely those =
in which the additional data can be deduced from the standard kernel and =
the base map>>
Indeed Kirill makes a double confusion (probably caused by the =
ellipticity of the redaction of my Note referenced below as [QGD]) about =
the meanings I attribute to the terms "extensor" and "regular".
As to the latter term, it has for me a purely smoothness meaning =
(referring to the fact that the equivalences on the manifolds have to be =
regular ones, and also that the anchor map of the kernel has to be a =
regular morphism, by which I mean composed of a surmersion and an =
embedding) and this term has to be dropped when dealing with the purely =
algebraic aspect of the question. In my paper referenced [MVF] I use =
instead (in view of more generality) the term "s-extensor", where the =
prefix "s-" refers to the more general context of "diptych" data (in =
which goodepics/good monos generalize and replace surmersions/embeddings =
; see my previous message), and, in the algebraic context, has to be =
read as just meaning "surjective" ( but is then considered as implied by =
the very term "extensor").
On the opposite the term "extensor" has for me a purely algebraic =
meaning (more restrictive than what is called "fibration" in Kirill's =
message) and is equivalent to the notion of what is called "regular =
fibration" in Kirill's message (where here "regular" is given an =
algebraic meaning which I never used for myself !), and refers to the =
very simple case, alluded to in my own previous message, where one can =
mimic exactly the theory of group extensions or surjective group =
homomorphisms (with the only caution of using two-sided cosets). The =
suffix "or" is to remind that this a funct-or instead of a funct-ion. =
(In the absence at the present day of any response from Marco, I guess, =
without being sure, that he was probably interested mainly in this case, =
or perhaps in the still more special case when the kernel reduces to a =
sum of groups ). It should be noticed that the (rather obvious) examples =
given at the end of my Note emphasize the independance of the algebraic =
conditions (extensor) and the smoothness ones (regular), so that the =
ambiguity, if it existed, should be cleared up for the careful reader =
(in the last of these examples the underlying algebraic functor is just =
the identity, but not the identity for the underlying smooth =
structures).
The so-called "fibrations" were out of the scope of [QFD], which was =
centred on the smoothness questions and not on the (obvious in that =
special case) algebraic aspect, implicitely (and perhaps imprudently !) =
considered there as "well known".
However they are considered, and play a basic role, in [MVF] (p. 238 =A7 =
7) under the name of "(surjective) exactors" (explained below), but the =
general problem of quotients and generalized kernels in the sense =
described and referenced by Kirill (which is of course much more =
delicate than the algebraically obvious case of extensors) is completely =
out of the scope of this paper. The remark (ii) of this page 238 =
emphasizes the fact that "extensor" implies "surjective exactor".
=20
Now it should be clear that when dealing with topological or smooth =
(i.e. Lie) groupoids, terms such as "(regular) fibrations" have to be =
definitely rejected, though previously used by various authors in the =
purely algebraic (or categorical) context, as giving rise to unsolvable =
ambiguities. (Note that there is a similar problem when using the =
widespread terminology "discrete" and "coarse" for groupoids ; though =
the ambiguity is generally much less disastrous in that case, I think =
it much better to use respectively the terms "null" and "banal"). The =
reason is of course that these terms have very ancient (various !) =
meanings in Topology and Differential Geometry, which are not at all =
implied by (nor imply) the algebraic condition, nor by weaker =
topological or smoothness conditions such as (surjective) open maps or =
surmersions.
=20
I remind that unhappily the term "foncteur fibrant" was introduced very =
early by Grothendieck and his school, and is, I think, still used by =
most category theorists, concurrently with the term "fibration", which =
appeared, I think, a little later. Ronnie Brown used also sometimes the =
more suggestive and non ambiguous term "star-surjective", which might =
become "star-surmersive" in the smooth case (and perhaps something like =
"star-epic" in my more genaral dyptich framework, though I don't intend =
to use it).
The terminology proposed in [MVF] (to which I refer for more precision =
and details omitted here) comes from a general analysis of the =
properties of a functor f between two groupoids, going from H to G =
(here we shall always assume below, to simplify, that f induces a =
surjective map for the bases, thus omitting as a consequence the word =
"surjective" in many occurences in what follows ; see [MVF] for more =
general and precise definitions and statements). Forgetting for a while =
(to make the things simpler) the smoothness (or diptych) framework to =
consider solely the algebraic properties, I believe that the most =
important of these are reflected by the two commutative squares a(f) and =
t(f) built, from f, respectively with the domain maps and the anchor =
maps (of H and G), and more precisely by the properties of =
injectivity/surjectivity/bijectivity of the two canonical arrows =
(denoted below by u and w) going from H to the pull back's generated by =
these two squares. (My general guess and philosophy is then that the =
suitable corresponding notions in the smooth or more generally "diptych" =
case -see my previous message- are gotten by just replacing =
"injective/surjective/bijective" by "good mono/good epi/iso", and that =
as soon as one is able to describe the set theoretic algebraic =
definitions, constructions and proofs by means of diagrams, everything =
extends "almost automatically" to the structured case, using the =
Godement diptych axioms).=20
In that context the extensors are just defined very quickly by the =
surjectivity of w and the "exactors" (here always surjective)by the =
surjectivity of u ("star-surjectivity" in the sense of Ronnie). (Note =
that the bijectivity of w characterizes the surjective equivalences, =
which are special instances of extensors).
Now it turns out readily that the bijectivity of u characterize those =
functors which describe actions of the groupoid G on the base of H (H is =
then called the action groupoid in the literature, but I emphasize the =
fact that the action of G is not fully described by H alone, but by the =
functor f). For that reason I believe quite natural (though I don't seem =
to be followed) to call "(surjective) actors" the functors of this type =
(note that the classical terminology in categorical works is "discrete =
fibrations" (!), "foncteurs d'hypermorphismes" (!!) in Ehresmann's =
book, and sometimes "star-bijective" for Ronnie Brown).=20
This explains (but perhaps does not justify) the above-mentioned term =
"exactor", with the suffix "or" as supra, and the prefix "ex" supposed =
to remind the surjectivity property of u (and not some terrorist or =
prejudicial activity) while evocating also some generalized kind of =
ex-tension.=20
The (surjective) actors and extensors appear as two opposite ways of =
degenerating for the (surjective) exactors, while the theory of Kirill =
and Philip explains how these two special cases are mixed up in the =
general (more sophisticated) case.
There is also a very interesting special case of exactors described in =
[MVF] under the name of "subactors" (Prop.-Def. 7.5), which are the =
faithful ones. They make up a subcategory whose arrows admit a unique =
factorization through a surjective equivalence and an actor. (As a =
general remark all the purely algebraic underlying content of [MVF] may =
be considered as more or less easy or even trivial and or more (or less =
?) well known, but again the interesting point is that the rather easy =
set-theoretical proofs may be (with some care) written diagrammatically =
in order to be transferred to the smooth case via the diptych method).
In a secund part, I'll add some other terminological remarks about the =
terminology of "quotient groupoids" used by Kirill.
=
Jean PRADINES
References (J. Pradines)
[QGD] Quotients de groupo=EFdes diff=E9rentiables, CRAS (Paris), t.303,
S=E9rie I, 1986, p.817-820.
[MVF] Morphisms between spaces of leaves viewed as fractions,
CTGDC (Cahiers de Topologie.....), vol.XXX-3 (1989),p. 229-246
----- Original Message -----
From: <K.Mackenzie@sheffield.ac.uk>
To: Categories List <categories@mta.ca>
Sent: Tuesday, July 01, 2003 11:33 AM
Subject: categories: quotients of groupoids
This is a comment on one aspect of Jean Pradines' very interesting
posting of June 8. I quote a large part of it for reference:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
.................................................................
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
In response to Jean's paper
[QGD] Quotients de groupo=EFdes diff=E9rentiables, CRAS (Paris), t.303,
S=E9rie I, 1986, p.817-820.
Philip Higgins and I wrote two papers ([HM90a] and [HM90b] below)
dealing with general quotients of Lie (=3Ddifferentiable) groupoids
and Lie algebroids. Our starting point was the idea that:
``a morphism in a given category is entitled to be called a quotient
map if and only if it is entirely determined by data on the domain''
(Perhaps this will seem naive to true categorists, but I write as an
end user of category theory, not a developer.)
I would summarize [QGD] as proving that the `regular extensors' are
quotient maps in the category of Lie groupoids. [HM90b] then showed,
by extending the notion of kernel, that in fact all extensors are
quotient maps in the category of Lie groupoids.
Philip and I used the term `fibration' for what Jean calls an
`extensor'; these maps satisfy a natural smooth version of the
notion of `fibration of groupoids' introduced by Ronnie Brown
in 1970 (building on work of Frolich, I think). (Throughout this
post, I assume base maps to be surjective submersions.)
There is unlikely to be a more general class of quotient maps for
Lie groupoids: the fibration condition on a groupoid morphism
$F : G \to H$ is exactly what is needed to ensure that any product
in the codomain is determined by a product in the domain: given
elements $h, h'$ of $H$ which are composable, one wants to be able
to write $h =3D F(g)$ and $h' =3D F(g')$ in such as way that $gg'$ will
exist and determine $hh'$; the fibration condition is the weakest
simple condition which ensures this.
Fibrations of Lie groupoids are not determined by their kernel in
the usual sense (=3D union of the kernels of the maps of vertex
groups); one also requires the kernel pair of the base map, and the
action of this (considered as a Lie groupoid) on the manifold of
one--sided cosets of the domain. This data, which Philip and I
called a `kernel system' is equivalent to a suitably well--behaved
congruence on the domain groupoid. Though more complicated than the
usual notion of kernel, the notion of normal subgroupoid system
gives an exact extension of the `First Isomorphism Theorem'.
We also showed that regular fibrations (=3D Jean's regular extensors)
are precisely those in which the additional data can be deduced from
the standard kernel and the base map; in the regular case the two
step quotient can be reduced to a single quotient consisting of
double cosets (as Jean remarks in his post). The congruences
corresponding to regular fibrations are those which, regarded as
double groupoids, satisfy a double source condition.
A good example of a fibration which is not regular is the division
map $\delta: (g,h)\mapsto gh^{-1}$ in a group $G$, considered as a
groupoid morphism from the pair groupoid $G\times G$ to the group $G$.
All of this (for Lie algebroids, as well as for Lie groupoids and
for vector bundles) is in the two papers referenced below. There is
also a full account coming in my book `General Theory of Lie
groupoids and Lie algebroids' (CUP) which should be appearing
in the next few months.
@ARTICLE{HM90a,
author =3D {P.~J. Higgins and K.~C.~H. Mackenzie},
title =3D {Algebraic constructions in the category of {L}ie
algebroids},
journal =3D {J.~Algebra},
year =3D 1990,
volume =3D 129,
pages =3D "194-230",
}
@ARTICLE{HM90b,
author =3D {P.~J. Higgins and K.~C.~H. Mackenzie},
title =3D {Fibrations and quotients of differentiable groupoids},
journal =3D {J.~London Math. Soc.~{\rm (2)}},
year =3D 1990,
volume =3D 42,
pages =3D "101-110",
}
Kirill Mackenzie
http://www.shef.ac.uk/~pm1kchm/
From rrosebru@mta.ca Mon Jul 7 11:44:01 2003 -0300
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Date: Mon, 7 Jul 2003 10:44:11 +0100
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From: Carron Shankland <carron@cs.stir.ac.uk>
Subject: categories: Announcement: AMAST 2004 (Stirling, July 2004)
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---------------------------------------------
CONFERENCE ANNOUNCEMENT
10th International Conference on
Algebraic Methodology And Software Technology
AMAST 2004
http://www.cs.stir.ac.uk/events/amast2004/
---------------------------------------------
July 12th - 16th, 2004
Stirling, Scotland, UK.
The major goal of the AMAST Conferences is to promote research that may
lead to the setting of software technology on a firm, mathematical basis.
This goal is achieved by a large international cooperation with
contributions from both academia and industry. The virtues of a software
technology developed on a mathematical basis have been envisioned as being
capable of providing software that is (a) correct, and the correctness can
be proved mathematically, (b) safe, so that it can be used in the
implementation of critical systems, (c) portable, i.e., independent of
computing platforms and language generations, and (d) evolutionary, i.e.,
it is self-adaptable and evolves with the problem domain. All previous
editions of the AMAST Conference, which were held at Iowa City (1989,1991),
Twente (1993), Montreal (1995), Munich (1996), Sydney (1997), Manaus
(1999), Iowa City (2000), and Reunion Island (2002), made contributions to
the AMAST goals by reporting and disseminating academic and industrial
achievements within the AMAST area of interest. During these meetings,
AMAST attracted an international following among researchers and
practitioners interested in software technology, programming methodology
and their algebraic and logical foundations. In addition, starting with the
1993 edition, the first day of each conference was dedicated to Mathematics
Education for Software Engineers.
TOPICS
------
As in previous years, we will invite papers reporting original research on
setting software technology on a firm mathematical basis. We expect two
kinds of submissions for this conference: technical papers and system
demonstrations. Of particular interest is research on using algebraic,
logic, and other formalisms suitable as foundations for software
technology, as well as software technologies developed by means of logic
and algebraic methodologies. Topics of interest include, but are not
limited to, the following:
SOFTWARE TECHNOLOGY:
* systems software technology
* application software technology
* concurrent and reactive systems
* formal methods in industrial software development
* formal techniques for software requirements, design
* evolutionary software/adaptive systems
PROGRAMMING METHODOLOGY:
* logic programming, functional programming, object paradigms
* constraint programming and concurrency
* program verification and transformation
* programming calculi
* specification languages and tools
* formal specification and development case studies
ALGEBRAIC AND LOGICAL FOUNDATIONS:
* logic, category theory, relation algebra, computational algebra
* algebraic foundations for languages and systems, coinduction
* theorem proving and logical frameworks for reasoning
* logics of programs
SYSTEMS AND TOOLS (for system demonstrations or ordinary papers):
* software development environments
* support for correct software development
* system support for reuse
* tools for prototyping
* component based software development tools
* validation and verification
* computer algebra systems
* theorem proving systems
IMPORTANT DATES
---------------
AMAST'2004 Conference: July 12-16, 2004
Final details have to be confirmed for paper submissions, but will most
likely be end December 2003/early January 2004.
PUBLICATION
-----------
Previous AMAST conferences have been published in the LNCS series by Springer.
LOCATION
--------
The conference will be held at the University of Stirling
http://www.stir.ac.uk/
CONTACT
-------
For further information, send email to amast@cs.stir.ac.uk
---------------------------------------------------------------------
Dr Carron Shankland Email: ces@cs.stir.ac.uk
Computing Science & Mathematics Tel: 01786-467444
University of Stirling http://www.cs.stir.ac.uk/~ces
Stirling FK9 4LA
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Hi all,
Does anybody know of an example of a Right Kan Extension of a functor
X:A->C along a functor F:A->B where B is not the 1 object category (as
this gives limits)? I'm looking for an intuitive example for a computer
scientist, not a mathematician (i.e. me!).
In "Categories and Computer Science" [Walters,1991], example 6 in
chapter 7 explains a situation in which the Left Kan Extension gives us
the arrows of the category B. As far as I can work out, the Right Kan
Extension for this set-up of A,B,C and functors X and F yields a trivial
answer. I am hoping to learn of a useful example to work with.
Regards,
Shane O'Conchuir
From rrosebru@mta.ca Mon Jul 7 15:18:25 2003 -0300
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Date: Mon, 07 Jul 2003 15:58:35 +0200
To: categories@mta.ca
From: Andree Ehresmann <Andree.Ehresmann@u-picardie.fr>
Subject: categories: Some history on quotient categories
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Here are some historical notes, in complement to Pradines' comments on=20
multiplicative graphs and on quotient categories.
Charles Ehresmann introduced multiplicative graphs as an algebraic support=
=20
to the notion of a "category kernel", which corresponds to the structure=20
induced on an open subset of a topological category (in the sense of=20
Charles, i.e. a category internal to Top). Such category kernels generalize=
=20
group kernels (that Charles used in the second part of his thesis to define=
=20
locally homogeneous spaces); we had considered them in the early sixties=20
both in his papers on differential geometry and in mine on optimization=20
problems. Then multiplicative graphs appeared as a natural tool in his=20
study on quotient categories.
Charles came to quotient categories by his works on foliations and on=20
prolongations of manifolds. In his long 1962 paper on foliations (1), he=20
constructs several kinds of holonomy groupoids by a quotient process. In=20
his paper on topological categories (2), he construct categories of jets=20
(local jets and infinitesimal jets) as quotient categories of the category=
=20
of local sections of a topological category. In fact he had exposed these=20
constructions much earlier in his lectures and briefly alluded to in his=20
papers on differential geometry in the late fifties.
So he was naturally led to a more formal study of quotients, which he began=
=20
in his paper "Structures quotient" (3), where in particular he introduces=20
the "strict quotient"; this paper (abstracted in (4)) is more easily read=20
than later papers. In a comment I have added to this paper in the "Oeuvres"=
=20
(comment 170, p. 375) I mentioned other authors who have studied quotient=20
categories about the same time, in particular Dedeckerand Mersch, Higgins,=
=20
Hoenke, P=FCmplun.
Later on, Charles tried to develop a non-abelian cohomology for which he=20
wanted to define a natural notion of "short sequence" in Cat. For this he=20
needed to define quotients of a category or of a groupoid by a=20
sub-category. It is done in the paper (5). The main results of this paper=20
are taken back in his book "Categories et Structures" (Dunod 1965), with=20
some illustrative diagrams (it is to this book that Pradines refers).
In several papers, he generalized the theory of quotient categories in the=
=20
frame of internal categories (cf. "Oeuvres", Parts III and IV), .
1. "Structures feuilletees", Proc. 5th Canadian Math. Congress; reprinted=
=20
in "Charles Ehresmann, Oeuvres completes et commentees" Part II-2, 563-626.
2. "Categories topologiques, III", Indig. Math. 28, 1966; reprinted in the=
=20
"Oeuvres" Part II-2, 655-669.
3. "Structures quotient",.Comm. Math. Helv. 38, 1963; reprinted in=20
"Oeuvres' III-1, 143-293.
4. "Structures quotient et cegories quotient", CRAS, 256, 1963, 5031-34,=20
reprinted in "Oeuvres" III-1, 9-11.
5. "Cohomologie dans une categorie dominee", Proc. Coll. Topologie, CBRM=20
1964; reprinted in "Oeuvres" III-2, 531-590.
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To: vardi@cs.rice.edu
Subject: categories: Call for Participation: LPAR'03 - September 22-26th, 2003, Almaty, Kazakhstan
Message-Id: <20030707183954.57ED64A9D1@cs.rice.edu>
Date: Mon, 7 Jul 2003 13:39:54 -0500 (CDT)
From: vardi@cs.rice.edu (Moshe Vardi)
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With apology for multiples copies.
=====================================================================
CALL FOR PARTICIPATION
LPAR 2003
September 22-26th, 2003, Almaty, Kazakhstan
http://www.lpar.net/2003/
International Conference on
Logic for
Programming
Artificial Intelligence and
Reasoning
PROGRAM CHAIRS
Moshe Y. Vardi
Department of Computer Science
Rice University
6100 S. Main St.
Houston, TX 77005, USA
Phone: +1-713-348-5977
Email: vardi@cs.rice.edu
Andrei Voronkov
Department of Computer Science
The University of Manchester
Oxford Rd.
Manchester M13 9PL, UK
Phone: +44-161-2756116
Email: voronkov@cs.man.ac.uk
TOPICS
* automated reasoning * description logics
* interactive theorem proving * nonmonotonic reasoning
* implementations of logic * specification using logics
* design of logical frameworks * logic in artificial intelligence
* program and system verification * lambda and combinatory calculi
* model checking * constructive logic and type theory
* rewriting * computional interpretations of logic
* logic programming * logical foundations of programming
* constraint programming * logical aspects of concurrency
* logic and databases * program extraction from proofs
* logic and computational complexity * modal and temporal logics
* translation validation * knowledge representation and reasoning
* proof-carrying code * reasoning about actions
* logic in semantic web * effectively presented structures
* proof planning
INVITED SPEAKERS
Franz Baader (TU Dresden): Automata and Tableaux methods for Description
and Modal Logics
Serikjan Badaev (Kazakh State National University: Computable Numberings
Dexter Kozen (Cornell University): TBA
Sergei Goncharov (Novosibirsk State University): TBA
Thomas Wilke (Christian-Albrechts University of Kiel): Minimizing
automata on infinite words.
IMPORTANT DATES
Early Registration: July 18, 2003
Conference: September 22-26, 2003
PROCEEDINGS
The proceedings will be published by Springer-Verlag in the LNAI
series and available at the conference.
ACCEPTED PAPAERS
Robert Nieuwenhuis and Albert Oliveras:
Congruence Closure with Integer Offsets
Dietmar Berwanger, Erich Graedel, Stephan Kreutzer:
Once upon a time in the west -- Determinacy, definability and complexity
of path games
Dietrich Kuske:
Is Cantor's theorem automatic?
Markus Lohrey:
Automatic Structures of Bounded Degree
Boris Konev, Anatoli Degtyarev, Michael Fisher:
Handling Equality in Monodic Temporal Resolution
Martin Fraenzle and Christian Herde:
Efficient SAT engines for concise logics: Accelerating proof search for
zero-one linear constraint systems
Matthias Baaz, Christian Fermueller:
A translation characterizing the constructive content of classical
theories
Juergen Giesl, Rene Thiemann, Peter Schneider-Kamp, Stephan Falke:
Improving Dependency Pairs
Sebastian Brandt, Anni-Yasmin Turhan, Ralf Kaesters:
Extensions of Non-standard Inferences to Description Logics with
transitive Roles
Serge Autexier, Carsten Schuermann:
Disproving False Conjectures
Barbara Morawska:
Completeness of E-unification with eager Variable Elimination
Davy Van Nieuwenborgh, Dirk Vermeir:
Ordered Diagnosis
Kumar Neeraj Verma:
On Closure under Complementation of Equational Tree Automata for Theories
Extending AC
F.J. Martin-Mateos, J.A. Alonso, M.J. Hidalgo, J.L. Ruiz-Reina:
A Formal Proof of Dickson's Lemma in ACL2
Toshiko Wakaki, Katsumi Inoue, Chiaki Sakama, Katsumi Nitta:
Computing Preferred Answer Sets in Answer Sets Programming
Furio Honsell, Marina Lenisa, Rekha Redamalla:
Strict Geometry of Interaction Graph Models
Margarita Korovina:
Fixed Points on Continuous Data Types
Silvio Ghilardi and Luigi Santocanale:
Algebraic and Model Theoretic Techniques for Fusion Decidability in
Modal Logics
Paola Bruscoli and Alessio Guglielmi:
On Structuring Proof Search for First Order Linear Logic
Quoc Bao Vo, Abhaya Nayak, Norman Foo:
A syntax-based approach to reasoning about action
Christoph Walther, Stephan Schweitzer:
A Machine-Verified Code Generator
Christoph Beierle, Gabriele Kern-Isberner:
A logical study on qualitative default reasoning with probabilities
Jean-Michel Couvreur, Nasser Saheb, Gregoire Sutre:
An Optimal Automata Approach to LTL Model Checking of Probabilistic Systems
D. Galmiche and J.M. Notin:
Connection-based proof construction in Non-Commutative Logic
Alberto Ciaffaglione, Luigi Liquori, Marino Miculan:
Imperative Object-based Calculi in (Co)Inductive Type Theories
Thierry Boy de la Tour, Mnacho Echenim:
NP-Completeness Results for Deductive Problems on Stratified Terms
Bernhard Heinemann:
Extended Canonicity of Certain Topological Properties of Set Spaces
ASSOCIATED WORKSHOP:
4th International Workshop on the Implementation of Logics
Saturday, September 27th, 2003, http://www.csc.liv.ac.uk/~konev/wil2003/
From rrosebru@mta.ca Tue Jul 8 12:18:58 2003 -0300
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To: categories@mta.ca
Subject: categories: Faked sender's address
Message-ID: <1057665607.3f0ab247668b3@inbox.math.yorku.ca>
Date: Tue, 08 Jul 2003 08:00:07 -0400 (EDT)
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[Note from moderator: Walter is not alone in having this experience, note
also that other subject lines are used, caveat lector.]
Dear categorist:
It seems that a number of categorists recently received a message showing as
sender tholen@mathstat.yorku.ca, as subject "Re: application", with an
attachment that, of course, one should not open. Let me clarify that I never
sent such a message. I have been told that in all likelihood my address was
used to disguise the virus-affected machine that sent the message and that was
the real target of the attack.
Regards,
Walter.
From rrosebru@mta.ca Wed Jul 9 11:12:43 2003 -0300
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From: Jpdonaly@aol.com
Message-ID: <116.25e2326c.2c3c7bad@aol.com>
Date: Tue, 8 Jul 2003 15:55:25 EDT
Subject: categories: Correction to terminology (Pat Donaly)
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Jean Pradines has kindly corrected my usage of "intertwining function". I
erred in attempting to extend this term to the variant naturality concept
described but not named by Mac Lane in problem 5, page 19 of either edition of
"Categories for the Working Mathematician". To within order of the functors which
are involved, this problem describes a natural transformation from functor F:A-->
B to parallel functor G:A-->B to be a function t which satisfies the left and
right transformation laws
t(ab)=G(a)t(b)=t(a)F(b)
when ab is defined in A. (The pair (G,F) is implicitly part of the natural
transformation.) If A is a group with object u and G and F are group actions,
then the value t(u) is a classical intertwining function, whereas I have been
calling t an intertwining function. I apologize to all of those who had any of
their time wasted in wondering about this connection.
I still need terminology for t much more than I need the intertwining
function idea; so, from now on, I intend to use "entwining function" to refer to t,
but, if this conflicts with more standard usage or if there is already a
standard word for t, I would appreciate being told.
Pat Donaly
From rrosebru@mta.ca Wed Jul 9 11:21:49 2003 -0300
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From: Jpdonaly@aol.com
Message-ID: <26.3c09a4ee.2c3cd0f3@aol.com>
Date: Tue, 8 Jul 2003 21:59:15 EDT
Subject: categories: Subcategories (Pat Donaly)
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Categorists:
Here is something which bothers me.
It seems to be common for textbook writers to prove that the composite of a
pair of monomorphisms is a monomorphism, and it may be taken for granted that
everyone knows that every object is a monomorphism. These two facts imply that
the monomorphisms in a category form a subcategory of it. A similar remark
applies to pullbacks: On page 16 of "Sheaves in Geometry and Logic", Mac Lane and
Moerdijk prove among other things that, in the category of commutative
squares of a category, the composite of a pair of pullbacks is a pullback, which is
a good start toward establishing that the pullbacks form a subcategory of the
commutative squares, but Mac Lane and Moerdijk are satisfied with calling the
multiplicativity of pullbacks a "pasting lemma" (the quotes are theirs). In
proposition 18.16. on page 121 of their 1973 book, "Category Theory", Herrlich
and Strecker do this sort of thing wholesale, leaving me to wonder, is there
something wrong with the subcategory concept?
To be honest, I have noticed that the habit of naming categories after
their objects to the extent possible makes it difficult to speak of a
subcategory which has the same objects as its parent, but it nevertheless
seems strange that, after generalizing the subobject notion, category
theory would terminologically orphan its own subobjects. Any
clarifications or corrections of these impressions?
Pat Donaly
From rrosebru@mta.ca Sun Jul 13 10:23:27 2003 -0300
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Date: Sat, 12 Jul 2003 19:35:10 +1000
To: categories@mta.ca
From: Ross Street <street@ics.mq.edu.au>
Subject: categories: Scott Russell Johnson Memorial Fellowship
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Please refer to the site http://www.ics.mq.edu.au/jobs/18743/ for
further information about the following position:
A Macquarie University Advertised Position:
DIVISION OF INFORMATION AND COMMUNICATION SCIENCES
Department of Mathematics
Research Fellow in Mathematics (Level A or B) (Scott Russell Johnson
Memorial Fellowship)
(Full-time (fixed-term)) Reference Number 18743
This Fellowship is funded by a generous donation to the Centre of
Australian Category Theory (CoACT) in memory of Dr Scott Russell
Johnson. The appointee will conduct research within CoACT. Other
duties include seminar presentations, postgraduate supervision and
relevant teaching. Applicants should indicate the level at which they
are applying, or whether they wish to be considered at both levels.
Essential Criteria at Level A: PhD (or submitted) in Mathematics or
related area or equivalent experience; strong background in a
relevant field of mathematics such as category theory, algebraic
topology, algebraic geometry, or low-dimensional topology; ability to
present research results at scientific meetings and to publish in the
scientific literature; excellent written and oral communication
skills; ability to work as a member of a team.
Additional Essential Criteria at Level B: At least three years
postdoctoral research experience in one of the above areas, or
equivalent; strong publication record; success in securing research
funding.
Enquiries: Professor Ross Street, Director of CoACT, on (02) 9850
8921 or e-mail
street@math.mq.edu.au
Application Package: http://www.ics.mq.edu.au/jobs or Elaine Vaughan
on phone +61 2 9850 8947, fax (02) 9850 8114 or e-mail
evaughan@math.mq.edu.au Selection criteria must be addressed in the
application.
The position is available from January 2004 on a full-time
(fixed-term) basis for a period of three years with the possibility
of further appointment subject to funding and performance.
Probationary conditions may apply.
Salary Range:
Level A - up to $62,581 pa, including base salary $39,098 to
$52,882 pa, up to 17% employer's superannuation and annual leave
loading; an appointee with a PhD will be appointed to a minimum of
Point 6 on the salary scale, currently $49,299 pa.
Level B - $65,845 to $78,079 pa, including base salary $55,640
to $65,978 pa, 17% employer's superannuation and annual leave loading.
Applications including full curriculum vitae, quoting the reference
number, visa status, and the names and addresses (including postal
and/or e-mail addresses) of three referees, should be forwarded to
the Recruitment Manager, Workplace Relations and Services, Macquarie
University, NSW 2109 by 22 August 2003. Applications will not be
acknowledged unless specifically requested.
The selection criteria of the position must be addressed in your application.
These positions are available to Australian residents, those who
hold valid working visas or permits, (Permanent Residency is required
for Continuing positions) or overseas applicants for senior positions.
Equal Employment Opportunity and No Smoking in the
Workplace are University policies.
From rrosebru@mta.ca Mon Jul 14 16:04:52 2003 -0300
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Date: Sun, 13 Jul 2003 19:34:48 +0200 (CEST)
From: Tom LEINSTER <leinster@ihes.fr>
To: categories@mta.ca
Subject: categories: Re: Compatibility of functors with limits
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A week ago I asked this list a question, thinking that lots of people
would know the answer. Either I was wrong or those who know are keeping
it to themselves, as I didn't get any answers at all. But I now
understand the issue better than before, so I'd like to try re-asking my
question in a different way and see if that elicits a response.
I'm trying to understand a certain notion of compatibility of a functor
with limits. There are of course several well-known such notions:
preserves, reflects, creates. I called mine (provisionally) "respecting"
limits. It is close to preservation, but not quite the same; it seems
more constructive and perhaps (dare I say it?) more natural.
Preservation is all very well when the domain category has all limits, or
all limits of whatever type we're concerned with. But otherwise, it seems
a bit suspect. For let F: A ---> B be a functor; preservation says that
given a diagram D: I ---> A in A,
- if D *does* admit a limit cone, then the image of that cone under F is
also a limit cone,
- if D *doesn't* admit a limit cone, then... well, nothing.
"Respect", on the other hand, says the same as preservation in the first
case, but also says something in the second case. (So respect is in
general stronger than preservation.)
Last time I gave a definition of respect in terms of categories of cones;
that definition is appended to this mail. Here's a different way to put
it: F "respects limits for D" if the canonical map
\int^a (\int_i A(a, Di)) \times Fa ----> \int_i FDi
is an isomorphism. Here \int^a denotes coend over a in A, \int_i is limit
over i in I, and I'm working under the assumption that these (co)limits
exist in the codomain category B. Now, if D does have a limit in A then
the left-hand side is
\int^a A(a, \int_i Di) \times Fa
which by density is just F \int_i Di; so, as claimed, respect is the same
as preservation in the case where the limit exists.
My question was whether anyone understood "respect of limits" well, or
could shed any light on it. It seems to me that, as well as being just
the right thing in certain examples I've been considering, it's a very
natural concept.
Tom
The definition of respect from last time:
> Let F: A ---> B be a functor, where B is a category with (for
> sake of argument) all small limits and colimits. Let I be a small
> category and D: I ---> A a diagram in A; write Cone(D) for the category
> of cones on/into D in A, write Cone(FD) for the category of cones on FD
> in B, and write
>
> F_*: Cone(D) ---> Cone(FD)
>
> for the induced functor. Then F can be said to "respect limits for D"
> if the colimit of F_* is the terminal object of Cone(FD) (that is, the
> limit cone on FD).
From rrosebru@mta.ca Mon Jul 14 16:06:19 2003 -0300
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Date: Sun, 13 Jul 2003 17:05:23 -0300 (BRT)
From: Flavio Leonardo Cavalcanti de Moura <flavio@mat.unb.br>
To: <categories@mta.ca>
Subject: categories: special elements and products.
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Let C and D be any two categories (probably small). Consider the category
that have all
functors from C to D as objects and natural transformations as morphisms.
Does this category have initial (or terminal) objects and binary products?
=09Best regards,
=09Fl=E1vio Leonardo.
From rrosebru@mta.ca Mon Jul 14 16:08:13 2003 -0300
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for categories-list@mta.ca; Mon, 14 Jul 2003 16:06:42 -0300
Message-Id: <200307140729.h6E7TUn18042@math-cl-n01.ucr.edu>
Subject: categories: chain complexes
To: categories@mta.ca (categories)
Date: Mon, 14 Jul 2003 00:29:29 -0700 (PDT)
From: "John Baez" <baez@math.ucr.edu>
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Dear Categorists -
Who first showed that an internal category in
the category of abelian groups was a 2-term chain
complex of abelian groups? What's a good reference?
Who first showed that an internal strict omega-category
in the category of abelian groups was a chain complex
of abelian groups? What's a good reference?
(Of course for "internal X in the category of Y's",
I am willing to accept "internal Y in the category of X's"
as a substitute.)
Best,
jb
From rrosebru@mta.ca Mon Jul 14 16:10:59 2003 -0300
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for categories-list@mta.ca; Mon, 14 Jul 2003 16:09:28 -0300
From: Jpdonaly@aol.com
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Date: Mon, 14 Jul 2003 14:47:48 EDT
Subject: categories: Groups vs. groupoids (Pat Donaly)
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To all category theorists:
While (partially) responding to Tom Leinster's 7/04 query regarding limit
preservation, a query of my own occurred to me: On pages 6 and 7 of Alain Connes'
book, "Noncommutative Geometry", he writes, "It is fashionable among
mathematicians to despise groupoids and to consider that only groups have an authentic
mathematical status, probably because of the pejorative suffix oid."
Professor Connes later cites the groupoid of states of the hydrogen atom in order to
eliminate the prejudice against groupoids, but, for group theorists, there is a
more direct way: Since Frobenius and/or Burnside adopted the concept of an
abstract group in order to consider general group actions and representations,
group theorists have been heavily involved in groupoids, whether they liked it
or not.
Is it generally understood by categorists that every group action---as a
comma category---is a groupoid?
Pat Donaly
From rrosebru@mta.ca Tue Jul 15 11:33:45 2003 -0300
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for categories-list@mta.ca; Tue, 15 Jul 2003 11:29:11 -0300
Message-Id: <200307151227.h6FCRDn26350@math-cl-n01.ucr.edu>
Subject: categories: Higher-Dimensional Algebra V: 2-Groups
To: categories@mta.ca (categories)
Date: Tue, 15 Jul 2003 05:27:13 -0700 (PDT)
From: "John Baez" <baez@math.ucr.edu>
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Here's a new paper that studies categorified groups and
Lie groups:
Higher-Dimensional Algebra V: 2-Groups
John C. Baez and Aaron D. Lauda
Abstract:
A 2-group is a "categorified" version of a group, in which the
underlying set G has been replaced by a category and the
multiplication map m: G x G -> G has been replaced by a functor.
Various versions of this notion have already been explored;
our goal here is to provide a detailed introduction to two,
which we call "weak" and "coherent" 2-groups. A weak 2-group
is a weak monoidal category in which every morphism has an
inverse and every object x has a "weak inverse": an object
y such that x tensor y and y tensor x are isomorphic to 1.
A coherent 2-group is a weak 2-group in which every object x
is equipped with a specified weak inverse x* and isomorphisms
i_x: 1 -> x tensor x*, e_x: x* tensor x -> 1 forming an
adjunction. We define 2-categories of weak and coherent
2-groups, construct an "improvement" 2-functor which turns
weak 2-groups into coherent ones, and prove this 2-functor
is a 2-equivalence of 2-categories. We also internalize
the concept of coherent 2-group, which gives a way to define
topological 2-groups, Lie 2-groups, affine 2-group schemes,
and the like. We conclude with a tour of examples.
Diagrammatic methods are emphasized throughout - especially
string diagrams.
This paper will soon appear on the mathematics arXiv, but
their computer seems unable to draw some of the pictures
correctly, so I urge you to try this PDF version instead:
http://math.ucr.edu/home/baez/hda5.pdf
The next paper in this series will study categorified Lie algebras.
From rrosebru@mta.ca Tue Jul 15 11:36:48 2003 -0300
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for categories-list@mta.ca; Tue, 15 Jul 2003 11:33:54 -0300
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From: "Mamuka Jibladze" <jib@rmi.acnet.ge>
To: <categories@mta.ca>
References: <Pine.LNX.4.44.0307131928140.12066-100000@ssh.ihes.fr>
Subject: categories: Re: Compatibility of functors with limits
Date: Tue, 15 Jul 2003 18:48:24 +0400
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It just occurred to me that there is something closely related
in lattice theory; unfortunately I cannot give a reference, but
I remember that one calls a subposet P' of a poset P
relatively (co)complete if whenever a subset of P' has an upper
bound in P, it has a least upper bound in P'.
A related question: does anybody know any analogs of the
Freyd's Adjoint Functor Theorems for functors between
in(co)complete categories?
Mamuka
From rrosebru@mta.ca Tue Jul 15 11:38:35 2003 -0300
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for categories-list@mta.ca; Tue, 15 Jul 2003 11:35:38 -0300
To: categories@mta.ca
Subject: categories: Inverse limits in Grothendieck categories
Reply-To: mhovey@wesleyan.edu
From: Mark Hovey <hovey@picard.math.wesleyan.edu>
Date: 15 Jul 2003 10:11:16 -0400
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It is pretty well-known that Grothendieck abelian categories have all
small limits. It is perhaps less well known that these inverse limits
do not have to have the same exactness properties as the usual module
inverse limits. For example, the infinite product is left exact, but
not exact, in a general Grothendieck category. Since Grothendieck
categories have enough injectives, one can take the right derived
functors of product and the right derived functors of inverse limit.
Since products are not exact, the inverse limit of a sequence could well
have infinitely many nonzero right derived functors, even if it satisfies
a Mittag-Leffler condition.
Does anyone know if right derived functors of products and inverse limits have
even been studied, either in general Grothendieck categories or in
specific examples?
Thanks,
Mark Hovey
From rrosebru@mta.ca Thu Jul 17 11:50:14 2003 -0300
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Date: Wed, 16 Jul 2003 19:22:02 +0200
To: categories@mta.ca
From: Andree Ehresmann <Andree.Ehresmann@u-picardie.fr>
Subject: categories: Re: Groups vs. groupoids (Pat Donaly)
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In answer to Pat Donaly
The connection between group actions and groupoids has been known and=20
extensively used for a long time. It was realized by Charles Ehresmann in=20
the early fifties. In fact Charles came to categories from groupoids, and=20
to groupoids from group actions and from pseudogroups of transformations.
In particular, in his works on fibre bundles and Differential Geometry, he=
=20
associated a groupoid to a pseudogroup of transformations (1), then=20
considered action of groupoids of jets as extending group actions (2).
In the paper (3) he introduces topological and differentiable categories=20
(i.e., internal to Top and to Diff), in view of associating to a principal=
=20
bundle H a particular topological groupoid P (called a locally trivial=20
groupoid). He then finds the locally trivial bundles associated to H as the=
=20
spaces on which there is an (internal) action of this groupoid. Given a=20
topological space F with an action of a sub-group of P, he constructs such=
=20
a space with fibre F by an "enlargement" process he had defined in his=20
important paper (4).
These results and many others can be found in the series of papers=20
reprinted in "Charles Ehresmann : Oeuvres completes et commentees" (more=20
specially in Part I), 1980-83..
(1) Les prolongements d'une vari=E9t=E9 diff=E9rentiable, Atti IV=20
Cong. dell'Unione Mate. Italiana, Taormina 1951, reprinted in "Oeuvres",=20
Part I, pp. 207-215.
(2) Introduction =E0 la th=E9orie des structures infinit=E9smales et des=20
pseudo-groupes de Lie, Actes Coll. Intern. Geom. Diff. Strasbourg, CNRS=20
1953, reprinted in "Oeuvres", Part I, pp. 217-230.
(3) Categories topologiques et categories differentiables, Coll. Geom.=20
Diff. Globale, CBRM Bruxelles 1959, reprinted in "Oeuvres", Part I, pp.=20
237-250.
(4) Gattungen von lokalen Strukturen, Jahres. d. Deutsches Math. 60-2,=20
1957, reprinted in "Oeuvres", Part II, pp. 125-153.
Andree C. Ehresmann
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Subject: categories: FM 2003 Call for Participation
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Sorry if some of you receive multiple copies of this message.
Stefania Gnesi (FM2003 General Chair)
==============================================================================
Call for Participation
------------------------------------------------------------------------------
The 12th International FME Symposium
Pisa, Italy - September 8-14, 2003
http://fme03.isti.cnr.it - fme03@isti.cnr.it
------------------------------------------------------------------------------
FM 2003 is the twelfth in a series of symposia organized by Formal
Methods Europe, an independent association whose aim is to stimulate
the use of, and research on, formal methods for software development.
These symposia have been notably successful in bringing together a
community of users, researchers, and developers of precise
mathematical methods for software development as well as industrial
users.
Formal methods have been controversial throughout their history, and
the realization of their full potential remains, in the eyes of many
practitioners, merely a promise. Have they been successful in
industry? If so, under which conditions? Has any progress been made
in dispelling the skepticism that surrounds them? Are they worth the
effort? Which aspects of formal methods have become so well
established in the industrial practices to loose the "formal method"
label in the meanwhile?
FM 2003 aims to answer these questions, by contributions not only
from the Formal Methods community but also from outsiders and even
from skeptical people who are most welcome to explain, document, and
motivate the source of their reluctance.
Satellite Events
FM 2003, will host 7 Workshops, 8 Tutorials and 1 Day dedicated to
the Industry besides the 3 days of the FME Symposium. Tool
demonstrations will also take place during the symposium, with the
opportunity of holding presentations for each tool.
For full details on the Symposium organization and to register please
see the web site http://fme03.isti.cnr.it, or send your query to
fme03@isti.cnr.it.
_______________________________________________
Mailinglist mailing list
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From rrosebru@mta.ca Thu Jul 17 11:53:59 2003 -0300
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for categories-list@mta.ca; Thu, 17 Jul 2003 11:53:50 -0300
From: Jpdonaly@aol.com
Message-ID: <1e.15927aee.2c470709@aol.com>
Date: Wed, 16 Jul 2003 15:52:41 EDT
Subject: categories: Generalized Yoneda Lemma (from Pat Donaly)
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To all category theorists:
Some time ago I noticed a generalization of the Yoneda Lemma which runs as
follows: As in the classical version, one begins with a nonvoid category A and a
pair of function-valued bifunctors on the product of the following
categories: The first is the category of function-valued natural transformations on A,
and the second is A itself. One of the bifunctors is the evaluation functor
(t,a)-->t(a), where t is a function-valued natural transformation on A, and a is
an A-morphism. Call this bifunctor E. The other bifunctor Y is described as
an iterated Yoneda embedding which has been unpartialled, and the classical
Yoneda Lemma states that Y is naturally isomorphic to E. Also, a particular
isomorphism is specified, at least on objects. But the situation is clarified by
observing that there is a bijective parameterization of the homset H of
natural transformations which entwine Y with E by what has to be called the center
of A, that is, by the commutative monoid C of natural transformations (under
(indifferently) pointwise or function composition) which entwine the identity
functor on A with itself. This parameterization sends natural isomorphisms to
natural isomorphisms, and, applying it to the identity functor on A gives the
classical Yoneda Lemma, but, of course, there might be other isomorphisms in C,
hence other isomorphisms of Y with E.
The parameterization formula from the center C into the homset H is not
particularly enlightening, but its inverse is literally the formula which expresses
an adjunctional unit in terms of its adjunction (regarded as a
function-valued natural bitransformation). In fact, this development all comes out of the
observation that the adjunctional unit formula extends to a concept which is
well beyond the idea of an adjunction, and this generalized unit concept even
includes naturality in a certain sense. Thus one nearly has naturality, the
Yoneda Lemma, the adjunctional unit concept and the idea of the center of a
category all together in one basket.
It is my personal opinion that these facts are most easily seen by working
with natural transformations in terms of their fully extended entwining
functions (so that "horizontal" composition is function composition and "vertical"
composition is pointwise composition), but ruthless experience tells me that
someone who is not familiar with the journal literature should take care not to
underestimate a priori the mysterious capabilities of diagrammers, who, after
all, created category theory in the first place. Moreover, I have no way to
assess the value of this generalization in terms of the classical applications of
the Yoneda Lemma, as in, say, the calculation of characteristic classes. Thus
I am interested in getting directions to relevant portions of the literature,
and would someone point out any applications of this generalized Yoneda
Lemma?
Pat Donaly
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for categories-list@mta.ca; Thu, 17 Jul 2003 11:59:47 -0300
Date: Wed, 16 Jul 2003 16:58:49 +0200 (CEST)
From: Tom LEINSTER <leinster@ihes.fr>
To: categories@mta.ca
Subject: categories: Re: Compatibility of functors with limits
In-Reply-To: <Pine.LNX.3.96.1030716154419.29520A-100000@plover.dpmms.cam.ac.uk>
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Mamuka Jibladze wrote:
> It just occurred to me that there is something closely related
> in lattice theory; unfortunately I cannot give a reference, but
> I remember that one calls a subposet P' of a poset P
> relatively (co)complete if whenever a subset of P' has an upper
> bound in P, it has a least upper bound in P'.
This is quite similar, but not the same. I'll take the dual concept (glbs
rather than lubs), since it was respect of limits that I wrote about
originally. Take an inclusion P' into P of posets, and take a diagram D
in P', which might as well be just a subset of P'. Then to say that the
inclusion of P' into P respects meets for D is to say that
join {lower bounds of D in P'}
= meet D
where both join and meet are taken in P. (I'm assuming that P is
complete; if not, respect of meets for D also asserts that the join and
the meet exist.) In my first mail I described, vaguely, respect of limits
as meaning that the limit of the image is "no bigger than it needs to be".
Order theory is (unsurprisingly) the context in which this makes the most
sense: the greatest lower bound of D in P obviously needs to be greater
than all the lower bounds of D in P', but that understood, it's minimal.
The dual of Mamuka's statement is that, with D and P' and P as above, if D
has a lower bound in P then it has a greatest lower bound in P'. Here's
an example where meets are respected but this condition (= relative
completeness?) fails. Let 0 be the empty category. For any category C,
the unique functor 0 ---> C respects limits if and only if C has an object
that is both initial and terminal. So if 1 is the one-element lattice
then 0 ---> 1 respects meets, and the subset 0 of 0 has a lower bound in 1
but no lower bound in 0.
Tom
From rrosebru@mta.ca Fri Jul 18 12:23:31 2003 -0300
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Date: Fri, 18 Jul 2003 14:39:52 +0200
To: categories@mta.ca
From: Marco Grandis <grandis@dima.unige.it>
Subject: categories: preprint: Normed combinatorial homology and noncommutative tori
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The following preprint is available in ps, at:
http://www.dima.unige.it/~grandis/Bsy2.ps
___________
Normed combinatorial homology and noncommutative tori
Marco Grandis
Abstract.
Cubical sets have a directed homology, studied in a previous paper
and consisting of preordered abelian groups, with a positive cone generated
by the structural cubes. By this additional information, cubical sets can
provide a sort of "noncommutative topology", agreeing with some results of
noncommutative geometry but lacking the metric aspects of C*-algebras.
Here, we make such similarity stricter by introducing *normed*
cubical sets and their *normed* directed homology, formed of normed
preordered abelian groups. The normed cubical sets associated with
"irrational rotations" have thus the same classification up to isomorphism
as the well-known irrational rotation C*-algebras.
MSC: 55U10, 81R60, 55Nxx.
Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology,
normed abelian groups.
Marco Grandis
Dipartimento di Matematica
Universita` di Genova
via Dodecaneso 35
16146 GENOVA, Italy
e-mail: grandis@dima.unige.it
tel: +39.010.353 6805 fax: +39.010.353 6752
http://www.dima.unige.it/~grandis/
From rrosebru@mta.ca Mon Jul 21 10:52:46 2003 -0300
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for categories-list@mta.ca; Mon, 21 Jul 2003 10:49:25 -0300
Message-Id: <200307200443.h6K4h5f04326@math-cl-n02.ucr.edu>
Subject: categories: Higher-Dimensional Algebra VI: Lie 2-Algebras
To: categories@mta.ca (categories)
Date: Sat, 19 Jul 2003 21:43:05 -0700 (PDT)
From: "John Baez" <baez@math.ucr.edu>
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Here's a new paper that studies categorified Lie algebras:
Higher-Dimensional Algebra VI: Lie 2-algebras
John C. Baez and Alissa S. Crans
The theory of Lie algebras can be categorified starting from a
new notion of "2-vector space", which we define as an internal
category in Vect. There is a 2-category Vect having these
2-vector spaces as objects, "linear functors" as morphisms and
"linear natural transformations" as 2-morphisms. We define a
"semistrict Lie 2-algebra" to be a 2-vector space L equipped
with a skew-symmetric bilinear functor satisfying the Jacobi
identity up to a linear natural transformation called the
"Jacobiator", which in turn must satisfy a certain law of its
own. This law is closely related to the Zamolodchikov tetrahedron
equation, and indeed we prove that any semistrict Lie 2-algebra
gives a solution of this equation, just as any Lie algebra gives
a solution of the Yang-Baxter equation. We construct a 2-category
of semistrict Lie 2-algebras and prove that it is 2-equivalent
to the 2-category of 2-term L-infinity algebras in the sense
of Stasheff. We also study strict and skeletal Lie 2-algebras,
obtaining the former from strict Lie 2-groups and using the
latter to classify Lie 2-algebras in terms of 3rd cohomology
classes in Lie algebra cohomology.
This paper will soon appear on the mathematics arXiv, but
the PDF version on my website looks a tiny bit better:
http://math.ucr.edu/home/baez/hda6.pdf
From rrosebru@mta.ca Mon Jul 21 10:53:56 2003 -0300
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Date: Sun, 20 Jul 2003 17:43:55 +0100
To: categories@mta.ca (categories)
From: Ronald Brown <ronnie@ll319dg.fsnet.co.uk>
Subject: categories: Re: chain complexes: in reply to John Baez
In-Reply-To: <200307140729.h6E7TUn18042@math-cl-n01.ucr.edu>
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reply to r.brown@bangor.ac.uk
The paper
(R. BROWN and HIGGINS, P.J.), `Cubical abelian groups with
connections are equivalent to chain complexes', Homology, Homotopy
and Applications, 5(1) (2003) 49-52.
has a reference to Grothendieck's `cat\'egorie
cofibr\'ee....' (1968) SLNM 79, (the canonical reference for the first
question),
and also to Bourn (JPAA 1990), and gives a proof that 5 different
structures are, in an additive category with kernels, equivalent to chain
complexes. Among these structures is strict globular omega-categories.
However, this is deduced from some non abelian and more difficult results.
Ronnie Brown
Dear Categorists -
Who first showed that an internal category in
the category of abelian groups was a 2-term chain
complex of abelian groups? What's a good reference?
Who first showed that an internal strict omega-category
in the category of abelian groups was a chain complex
of abelian groups? What's a good reference?
(Of course for "internal X in the category of Y's",
I am willing to accept "internal Y in the category of X's"
as a substitute.)
Best,
jb
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Date: Sun, 20 Jul 2003 19:50:35 +0200 (CEST)
From: Tom Leinster <leinster@ihes.fr>
To: categories@mta.ca
Subject: categories: Re: Compatibility of functors with limits
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Mamuka Jibladze wrote:
> A related question: does anybody know any analogs of the
> Freyd's Adjoint Functor Theorems for functors between
> in(co)complete categories?
Borceux states the 'More General Adjoint Functor Theorem' in Vol 1, 6.6.1
of his Handbook. This requires only that the codomain of the hoped-for
left adjoint is Cauchy-complete (and of course that the known functor has
some properties: it is 'absolutely flat' and satisfies some solution set
conditions).
Here's a representability theorem, presumably related. Let C be a small,
Cauchy-complete category and let X: C ---> Set. Then
X is representable <=> X respects small limits.
The same goes for familial representability and connected limits. Proofs
are at
http://www.ihes.fr/~leinster/rr.ps
Tom
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Date: Mon, 21 Jul 2003 13:16:32 -0700 (PDT)
From: John MacDonald <johnm@math.ubc.ca>
To: categories@mta.ca
cc: John MacDonald <johnm@math.ubc.ca>
Subject: categories: CT04
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Please circulate to all interested colleagues:
SECOND ANNOUNCEMENT
INTERNATIONAL CATEGORY THEORY CONFERENCE (CT04)
July 18-24, 2004
University of British Columbia
Vancouver, Canada
This conference will be held on the University of British Columbia campus.
It will begin with a reception at 6pm on Sunday July 18, 2004, and will
end at 1pm on Saturday July 24, 2004. All those interested in category
theory and its applications are welcome.
The CT04 website is http://www.pims.math.ca/science/2004/CT04
Currently only the link to Visitor Information has been activated. This
contains information which may be helpful when making travel plans.
Links to Registration and Accommodation will be activated during
September 2003. A further announcement will be made when these links
are active.
This conference is being organized with the help of the Pacific
Institute of Mathematics(PIMS}.
John MacDonald
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Date: Wed, 23 Jul 2003 08:32:11 +0200
To: categories@mta.ca
From: grandis@dima.unige.it (Marco Grandis)
Subject: categories: preprint: Normed combinatorial homology and noncommutative tori
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The following preprint is available in ps:
Normed combinatorial homology and noncommutative tori
Marco Grandis
Dip. Mat. Univ. Genova, Preprint 484 (Jul 2003), 14 p.
http://www.dima.unige.it/~grandis/Bsy2.ps
Abstract.
Cubical sets have a directed homology, studied in a previous paper
and consisting of preordered abelian groups, with a positive cone generated
by the structural cubes. By this additional information, cubical sets can
provide a sort of "noncommutative topology", agreeing with some results of
noncommutative geometry but lacking the metric aspects of C*-algebras.
Here, we make such similarity stricter by introducing *normed*
cubical sets and their *normed* directed homology, formed of normed
preordered abelian groups. The normed cubical sets associated with
"irrational rotations" have thus the same classification up to isomorphism
as the well-known irrational rotation C*-algebras.
MSC: 55U10, 81R60, 55Nxx.
Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology,
normed abelian groups.
Marco Grandis
Dipartimento di Matematica
Universita` di Genova
via Dodecaneso 35
16146 GENOVA, Italy
e-mail: grandis@dima.unige.it
tel: +39.010.353 6805 fax: +39.010.353 6752
http://www.dima.unige.it/~grandis/
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Date: Sun, 27 Jul 2003 11:41:37 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>
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To: categories@mta.ca
Subject: categories: Louis Nel in the news
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On a recent warm summer's Monday evening, I met Louis Nel, club
founder and current president for a game and plenty of instruction.
* * *
"It's a game with a magical touch," says Nel, a retired Professor of
Mathematics at Carleton University with a PhD from Cambridge. "When
I first saw the game, I was enchanted. My reaction was very similar
to when I fell in love. There's a real kinship between croquet
players who feel this. And it is definitely a different experience
than meeting another toothbrush user."
When he started playing croquet, Nel "knew nothing about the game at
all. For me it was just an image out of Alice in Wonderland." But
when the soft-spoken South African -- who is now a Team Canada
player and ranked 18th in North America -- bought a toy croquet set
for his four children from Canadian Tire, he was hooked. "The
children moved on and thought of it as nothing more than a good
game, but I saw that there was a lot more to croquet than a toy set.
So I went to nearly every sports store in Ottawa and no one knew
anything about the game."
Nel pursued his passion for a real croquet set and once he'd found
one he drove all the way to Toronto just to look at it. He didn't
even buy it. But when he did finally buy a set, it had a copy of a
croquet magazine in it and that was the lead he needed that set him
on the path to founding Croquet Ottawa.
For a year, Nel drove to Montreal once or twice a week to play at a
Westmount club, but he realized that it was no long-term solution.
He needed a club nearby and he realized that they only way to get it
was to create it himself.
* * *
Nel tells the story of a bunch of lawyers who came to play at the
club for an office outing. As one young man lined up with a glint in
his eye, he hit his opponent's ball to the far side of the court.
"So sorry, Boss," he said with a huge grin. This captures the
essence of the game. Croquet can be highly competitive and really
quite nasty, but at the same time it's lots of fun.
Copyright 2003 CanWest Interactive
Ottawa Citizen
July 26, 2003 Saturday Final Edition
SECTION: The Citizen's Weekly: Style; Pg. E3
LENGTH: 1391 words
HEADLINE: Serenity on the green: Croquet is all grown up and making
its presence felt in Ottawa
SOURCE: The Ottawa Citizen
BYLINE: Hattie Klotz
BODY: At the noisy corner of Bronson Avenue and Gladstone Street,
where trucks thunder by and a film of city grime and street dirt
covers almost everything, there's an oasis of calm. Sparkling like an
emerald in the dust, the smooth, green lawns of the Central Lawn
Bowling club form a square, bounded by chain-link fences, at this busy
intersection. It's here that Croquet Ottawa makes it home.
Croquet, you say? Yes, that childhood game that entails hitting a ball
through a hoop, once played on lawns and at cottages across the
country, has some serious players right here in Ottawa. While the
rules of the game vary according to geography (there's a North
American version, an international version, a version that is most
popular in Egypt, a cottage version), three times a week, the members
of Croquet Ottawa gather at the club on Gladstone or at Elmdale Lawn
Bowling club where they also have playing privileges, to enjoy a game
or two.
Croquet is the perfect way to pass a couple of hours. On a recent warm
summer's Monday evening, I met Louis Nel, club founder and current
president for a game and plenty of instruction. As soon as club
members measured out the correct distances on the court -- croquet is
nothing if not a game of precision -- and hammered in the hoops, two
games got underway side-by-side.
We played Golf Croquet, a version of the game that was new to me, but
proved simple and easy to grasp. There were plenty of exciting moments
during our fast and fluid game. And I found that as I concentrated on
each of my plays and their consequences several turns down the line,
everything but the game faded into the background. I didn't notice the
noise from the street. I couldn't imagine that I was in the middle of
the city. It was enough to be simply playing croquet.
The sound of mallet on ball and the satisfying thud when struck
cleanly and correctly, absorbed my attention. I realized croquet is
more than a game for children and elderly aunts. It is a game of
strategy. It is a mental game, somewhat like a cross between snooker
and croquet. It's a game for the soul too, as there's a serene,
timeless quality about playing as the sun sets and the lights come on
around this small square of green in the middle of so much concrete.
There's a certain contemplative side to croquet, since it's not
athletically demanding. And it's always played in beautiful
surroundings on velvet-smooth lawns.
"It's a game with a magical touch," says Nel, a retired Professor of
Mathematics at Carleton University with a PhD from Cambridge. "When I
first saw the game, I was enchanted. My reaction was very similar to
when I fell in love. There's a real kinship between croquet players
who feel this. And it is definitely a different experience than
meeting another toothbrush user."
When he started playing croquet, Nel "knew nothing about the game at
all. For me it was just an image out of Alice in Wonderland." But when
the soft-spoken South African -- who is now a Team Canada player and
ranked 18th in North America -- bought a toy croquet set for his four
children from Canadian Tire, he was hooked. "The children moved on and
thought of it as nothing more than a good game, but I saw that there
was a lot more to croquet than a toy set. So I went to nearly every
sports store in Ottawa and no one knew anything about the game."
Nel pursued his passion for a real croquet set and once he'd found one
he drove all the way to Toronto just to look at it. He didn't even buy
it. But when he did finally buy a set, it had a copy of a croquet
magazine in it and that was the lead he needed that set him on the
path to founding Croquet Ottawa.
For a year, Nel drove to Montreal once or twice a week to play at a
Westmount club, but he realized that it was no long-term solution. He
needed a club nearby and he realized that they only way to get it was
to create it himself.
Croquet Ottawa was formed in 1994 with founding members Ken Shipley, a
co-team Canada member who stills plays at the club -- his Team Canada
sign remains above the door in the clubhouse -- and Robert Armstrong,
who has now moved to Northern California, "but who gave me the nudge
of encouragement and was a constant spiritual presence and
enthusiasm," says Nel. "When you're starting something like this, you
need some luck and for me, Ken has been a real shot in the arm.
Another is Dean Chamberlain, a very gregarious man who has recruited
many players." Chamberlain is vice-president of the club and Shipley
began playing 10 years ago when he played at a garden party and, "had
bad knees from tennis so was looking for something else to do," he
says.
Croquet Ottawa now counts approximately 10 regular players among its
members. They make up a snapshot of Canadian demographics. There are a
couple of Canadians, a South African, an Englishman, a Ukrainian and
an Italian.
The evening I played, our foursome consisted of me, a Brit, Stuart, a
Canadian, Louis, South African and Marco, an Italian. Only in Canada
could this happen. They don't play croquet in Italy. "I was looking
for a summer sport," explained Marco, "as I snowboard in the winter
and I just didn't know what to do. A friend suggested I try this and
..." Well, four years later, Marco plays at a competitive level, with
a glint in his eye and a killer instinct. But in spite of the
opportunity for really vindictive and venomous play that croquet
presents -- it's all too tempting to hit another player's ball for six
-- "everybody is very relaxed and at peace in the club," says Nel,
"and we've never had a moment of strife."
Nel tells the story of a bunch of lawyers who came to play at the club
for an office outing. As one young man lined up with a glint in his
eye, he hit his opponent's ball to the far side of the court. "So
sorry, Boss," he said with a huge grin. This captures the essence of
the game. Croquet can be highly competitive and really quite nasty,
but at the same time it's lots of fun.
Before the birth of Croquet Ottawa, there were other traces of the
game in the city, "principally on the lawns of private homes in
Rockcliffe," says Nel. But there's also a croquet court in Rockcliffe
Park. Directly opposite Ashbury School, the lawn is a hidden gem.
Enclosed by eight-foot tall cedar hedges, it has a newly refurbished
clubhouse, flowerbeds overflowing with day lilies, picnic tables and
benches under fruit trees and scattered pieces of stone sculpture.
"It used to be a lawn bowling club," says Brian Murray, "that saw it's
heyday in the 1950s and '60s. It was a very busy part of Rockcliffe
social life. But it became primarily a croquet club in the 1980s," due
to the enthusiasm of Patrick Murray, former Mayor of Rockcliffe Park.
"He had the lawn specially seeded and mowed. We measured it out to
international standards and brought in experts from Saratoga, New York
and West Palm Beach in Florida. We played tournaments with these clubs
and we used to have a diplomatic tournament. There's even a huge
trophy. It's a giant croquet mallet, about 10 feet tall."
But with the amalgamation of Rockcliffe with the City of Ottawa two
years ago, the club died. "The minute Rockcliffe lost its autonomy,
the city really didn't have the funds to put into the upkeep of the
lawn," says Murray. Now, during the winter the land is turned into a
skating rink for children, which means that the grass is destroyed
come spring.
Croquet is a game that is dear to many people's hearts. It's easy to
grasp and can be played at many levels. It's a welcoming game and so
are the members of Croquet Ottawa. While we're playing, a man and a
small boy stop their bikes and peer through the fence. "What are you
doing?" chirps the boy. "We're playing croquet," someone replies. "Ah,
yes," sighs the man, "I used to play when I was a child." A couple of
minutes later the pair have cycled around the fence to the club gate.
They lean their bikes up against a bench and watch for a moment.
Within minutes, Ken Shipley has given the pair a hoop, a mallet and a
brightly coloured ball and they're practising happily at the end of
the court.
Croquet Ottawa is holding an Association tournament at the Elmdale
Lawn Bowling Club on July 26 and 27. Information on the club can be
found at http://www.magma.ca/ [tilde] acna/co.htm
Hattie Klotz is an Ottawa writer.
GRAPHIC: Photo: Chris Mikula, For Style Weekly; Canadian champion
croquet player and president of Croquet Ottawa, Louis Nel lines up a
difficult shot. 'It's a game with a magical touch,' he says.; Photo:
Chris Mikula, For Style Weekly; Canadian champion croquet player and
president of Croquet Ottawa, (Louis Nel) lines up a difficult shot.
'It's a game with a magical touch,' he says.
From rrosebru@mta.ca Mon Jul 28 11:21:59 2003 -0300
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Date: Mon, 28 Jul 2003 13:05:37 +0200
Message-Id: <200307281105.h6SB5bK19249@pc157aa.liacs.nl>
To: categories@mta.ca
Subject: categories: Second International Symposium on Formal Methods for Components and Objects
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(We apologize for the reception of multiple copies)
*********************** CALL FOR PARTICIPATION **********************
Second International Symposium on
Formal Methods for Components and Objects
(FMCO 2003)
DATES 4 - 7 November 2003
PLACE Lorentz Center, Leiden University, Leiden, The Netherlands
URL http://fmco.liacs.nl/fmco03.html
OBJECTIVES
The objective of this symposium is to bring together researchers and
practioners in the areas of software engineering and formal methods to
discuss the concepts of reusability and modifiability in
component-based and object-oriented software systems.
FORMAT
The symposium is a four days event in the style of the former REX
workshops, organised to provide an atmosphere that fosters
collaborative work, discussions and interaction. The program consists
of keynote and technical presentations, and contains an exquisite
social event. Speakers' contributions will be published after the
symposium in Lecture Notes in Computer Science by Springer-Verlag.
KEYNOTE SPEAKERS
Desmond D'Souza (Kinetium, Austin, USA)
E. Allen Emerson (University of Texas at Austin, USA)
Andrew D. Gordon (Microsoft Research, UK)
Yuri Gurevich (Microsoft Research, USA)
Tony Hoare (Microsoft Research, UK)
David Parnas (University of Limerick, IE)
Joseph Sifakis (Verimag, FR)
TECHNICAL PRESENTATIONS
Albert Benveniste (IRISA/INRIA - Rennes, FR)
Frank de Boer (CWI, NL)
Egon Boerger (Pisa University, IT)
Werner Damm (University of Oldenburg, DE)
Razvan Diaconescu (IMAR, RO)
Gregor Engels (University of Paderborn, DE)
Jose Luiz Fiadeiro (University of Leicester, UK)
Jan Friso Groote (Eindhoven University of Technology, NL)
Jean-Marc Jezequel (IRISA, Rennes, FR)
Bengt Jonsson (Uppsala University, SE)
Yassine Lakhnech (University of Grenoble, FR)
Rob van Ommering (Philips Research Laboratories, NL)
Amir Pnueli (The Weizmann Institute of Science, ISR)
Willem-Paul de Roever (University of Kiel, DE)
Jan Rutten (CWI, Amsterdam, NL)
Philippe Schnoebelen (CNRS, Cachan, FR)
Natalia Sidorova (Eindhoven University of Technology, NL)
Heike Wehrheim (University of Oldenburg, DE)
Jeannette Wing (Carnegie Mellon University, USA)
REGISTRATION
Participation is limited to about 80 people, using a first-in
first-served policy. To register, please fill in the registration
form at http://fmco.liacs.nl/fmco03.html. The EARLY registration fee
(BEFORE September 15, 2003) is 375 euro for regular participants and
250 euro for students It includes the participation to the symposium,
a copy of the proceedings, all lunches and refreshments, and a social
event (with dinner).
ORGANIZING COMMITTEE
F.S. de Boer (CWI and Utrecht University)
M.M. Bonsangue (LIACS-Leiden University)
S. Graf (Verimag)
W.P. de Roever (CAU)
For more information about participation and registration see the FMCO
site above or consult either F.S. de Boer (frb@cwi.nl) or
M.M. Bonsangue (marcello@liacs.nl).
From rrosebru@mta.ca Tue Jul 29 12:35:15 2003 -0300
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Subject: categories: CMS Summer 2004
To: categories@mta.ca
Date: Tue, 29 Jul 2003 10:19:31 -0300 (ADT)
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The Summer 2004 Meeting of the Canadian Mathematical Society will
be held at Dalhousie University, Halifax, Canada from Sunday, June
13 to Tuesday, June 15. Readers of this list may be interested in
some of the research sessions of the meeting:
Topos Theory, Organizer: Myles Tierney (Rutgers)
Hopf Algebras and Related Topics, Organizer: Yuri Bahturin (Memorial)
Algebraic Topology, Organizers: Keith Johnson (Dalhousie) and
Renzo Piccinini (Milan)
RJ Wood
Summer 2004 Meeting Director
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Date: Wed, 30 Jul 2003 11:09:16 +0200
Subject: categories: Conference announcement - Ramifications of Category Theory
From: alberto peruzzi <albertperuzzi@libero.it>
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Florence, Italy
November 18-22, 2003
University of Florence
Ramifications of Category Theory
A Workshop
including a featured lecture series by F. W. Lawvere
A Workshop on the Ramifications of Category Theory will take place in
Florence, Italy, from November 18 to November 22, 2003.
The workshop will pay attention not only to issues internal to category
theory, and in particular to topos theory, but also to manifold
relationships of category theory with topology, logic, philosophy, physics
and theoretical computer science, as witnessed by the list of invited
speakers. The aim is that of providing a recognition of present-day
frontiers of category-theoretic research and a perspective on future
research in the field, with emphasis on the foundations of mathematics and
the applications of category theory.
The workshop is also intended to honor Professor F.W. Lawvere on the 40th
anniversary of his Doctoral thesis on Functorial Semantics for Algebraic
Theories and to help bring into clearer focus the philosophical consequences
of his work, in particular the deep unification of geometry, algebra and
logic and of the concepts and tools of these areas of mathematics which have
stemmed from it.
Professor Lawvere will give a series of lectures, one central topic of which
will be quality, in its relation to space and quantity. Intensive and
extensive aspects of quality, together with an adjoint characterization of
it will be examined.
The general goal of his lectures will be to concentrate some
essentials from research in algebra and geometry to arrive at precise
philosophical formulations as a guide to the pursuit of learning and
future research in mathematical sciences.
For further information (in English and in Italian)
Visit the web page under construction:
http://ramcat.scform.unifi.it/
Alberto Peruzzi
Dipartimento di Filosofia
Via Bolognese 52
50139 Firenze
Italia
alper@unifi.it
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Reply-To: <noson@sci.brooklyn.cuny.edu>
From: "Noson Yanofsky" <noson@sci.brooklyn.cuny.edu>
To: "categories@mta. ca" <categories@mta.ca>
Subject: categories: Godel and Bernays on CT.
Date: Wed, 30 Jul 2003 12:34:42 -0400
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Recently Volume 4 of Kurt Godel=92s Collected Works came out.
Volume 4 and 5 are correspondences. There is some discussion of
category theory that I thought might be of interest. The relevant
portions have been extracted. First is Godel=92s letter to Bernays.
Then Bernays=92 response. Finally there is Solomon Feferman
discussion.
Letter 47. Godel to Bernays (9 January 1963) on page 221:
=93... I found it interesting that you speak on P. 199 of the
=93newer abstract disciplines of mathematics=94 as something lying
outside of set theory. I conjecture that you are thereby alluding
to the concept of category and to the self-applicability of
categories. But it seems to me that all of this is contained
within a set theory with a finitely iterated notion of class,
where reflexivity results automatically through a =93typical
ambiguity=94 of statements. Isn=92t that also your opinion? I=92ve
heard, by the way, that someone has formulated the axioms of set
theory with the aid of the concept of category and that this has
perhaps even been published. If you know something about it, I
would be very grateful to you for relevant information.=94
Letter 48. Bernays to Godel (23 February 1963) on pages 229-231:
=93That in the enumeration of the domains of mathematics in which
the classical methods come to be used, I named the =93newer
abstract disciplines=94 in addition to analysis and set theory was
conceived in the sense of a distinction between categorical and
hypothetical mathematics. Abstract axiomatic topology and algebra
can be pursued in such a way that one indeed uses concepts like
that of natural number and real number as well as set-theoretical
concepts and theorems, without undertaking a strict incorporation
into set theory.
As to the difficulties associate with =93categories=94, of which Mr.
Mac Lane spoke in his Warsaw address, I have only a rough idea of
the requirements in question. All the same, it seems to me that
the difficulties rest at least in part on the fact that it is
exclusively the extensional characterization of categories that
is being considered. In an analogous way, for example, if we
characterize the number 5 by means of the class of five-element
sets, there is an impediment to forming a class with the number 5
as [an] element. To be sure, categories are also effectively
given by means of axiom systems, and the multiplicity of the
axiom systems that have to be considered can presumably be
represented as a set, and certainly as a class. Your idea that a
set theory with a finitely iterated notion of class is suitable
as a framework for the theory of categories seems very plausible
to me. If axiomatic set theory is being extended in any event,
then the other novel infinitistic investigations should also find
a place in the extended framework. Just recently I received a
manuscript of the dissertation submitted ny William Hanf, in
which the author expresses the opinion in an introductory chapter
that for his investigations a set theory with a finitely iterated
notion of class does not suffice as an axiomatic framework.
=93That someone recently formulated the axioms of set theory with
the help of the notion of category I learned for the first time
from your letter.=94
This is Solomon Feferman=92s discussion of the correspondence on
pages 59-60:
=93Foundations of category theory. The foundational aspects of the
subject of category theory came up for discussion in Godel=92s
letter of 9 January 1963 (number 47); his point of departure was
a remark in Bernays 1961a to the effect that =93the =91newer=92
abstract disciplines of mathematics=92 [are] something lying
outside of set theory.=94 Godel assumed (mistakenly as it turned
out) that Bernays was =93thereby alluding to the concept of
category and to the self-applicability of categories=94. With
reference to self-applicability, what Godel presumably had in
mind are examples such as the category of all categories, for
which there is no straightforward set-theoretical interpretation.
He went on, interestingly, to suggest that such cases of
self-reference could be handled through =93typical ambiguity=94
applied to an extension of set theory by classes of finite type.
Actually, something like that had been pursued in the
Grothendieck school of homological algebra (e.g., in Gabriel
1962), which assumes 9instead of higher types) the existence of
arbitrarily many =93universes=94, i.e., collections of sets closed
under various standard operations, such as the stages Va in the
cumulative hierarchy for a strongly inaccessible. But for typical
ambiguity, one would need that the properties (in the language of
set theory, or higher type theory as suggested by Godel) of any
universe used are the same as in any other universe. Kreisel
suggested in 1965, pp.117-118, the required applications could
just as well be taken care of by use of the reflection principle
in set theory without the assumption of inaccessible cardinals;
the idea was spelled out and verified to a considerable extent in
Feferman 1969, though not all the problems were dealt with
thereby.
=93Also in letter 47, Godel said that he had heard =93that someone
has formulated the axioms of set theory with the aid of the
concept of category and that this has perhaps even been
published,=94 and he asked Bernays if he knew anything about it.
The first publication of an axiomatic theory of the category of
sets was by F. William Lawvere in his 1964; since that appeared
in the Proceedings of the National Academy of Sciences, it is
possible that Godel had heard of its submission through one of
the other members of the academy. In any case, Bernays=92 response
in letter 48 was not really helpful in dealing further with
either question concerning category theory, and the subject was
not pursued further in the correspondence.=94