Category Theory Yesterday Today (and Tomorrow?) : A Colloquium in Honour of Jean Benabou

A Colloquium : “Category Theory : Yesterday, Today and Tomorrow” sponsored by The Archive Trust for Mathematical Sciences and Philosophy together with The Categories en Fondements Programme of The Universite Paris 7 Denis Diderot and the Institut Mathematiques de Jussieu took place in The Salle Henri Cartan at The Ecole Normale Superieure Paris on Friday 3rd and Saturday 4th June 2011.

This event was intended to honour Professor Jean Benabou as he entered his 80th Year and to be a reflection and celebration of his many contributions, in particular to the Theory of Categories, during an active and continuing research career already spanning over 50 years.

The Colloquium was  preceded by a series of Rencontres/Discussions between Professsor Benabou and a panel of his mathematical colleagues which which took place between the afternoon of Monday 30th May and the morning of Friday 3rd June. Attendance at these Rencontres was by invitation.

The Agenda for these Rencontres/Discussions will shortly be added to this page and edited Transcripts of the Discussions and extracts from the recordings will also be made available here in due course.

Important Acknowledgement : The Archive Trust and The Colloquium Organiser wish to offer particular thanks and grateful acknowledgement to The Seminaire de Logique Categorique at the Institut Mathematique Jussieu and in particular to  Dr Paul-Andre MELLIES, without whose indispensable aid and support – both financial and adminsitrative – the Colloquium would not have been possible.

THE PROGRAMME of The Colloquium was as follows

VENUE : Salle Henri Cartan, Department of Mathematics, Ecole Normale Superieure 45 Rue D’Ulm, Paris 75005

Links to the recordings of the Talks below and to notesof the speakers presentations (or copies of overheads where used) will be operational on this page shortly.

Further Note : Professor Francois CHARGOIS (Universite Nancy 2, Archives Henri Poincare) and Professor Christian HOUZEL (CNRS) were prevented by illness from speaking at the Colloque. The Text of Professor CHARGOIS’s Talk entitled “BOURBAKI, GROTHENDIECK ET LA THEORIE DES CATEGORIES” was read into the Record of the Colloquium and will appear in the online record of the Proceedings below in due course.

We wish both Professor Chargois and Professor Houzel a swift and complete recovery.

FRIDAY 3rd June 2011

9:30 AM : Welcome and Introductory Remarks

9:45 – 10:45 AM :

Professor Pierre CARTIER (IHES, CNRS) :

TITLE :  GROUPOIDES HIER ET AUJOURD HUI

RESUME : Les groupoïdes ont été inventés par Brandt vers  1930 , à l’occasion d’un problème d’arithmétique : généralisation de la théorie des idéaux de Dedekind au cas non commutatif . Prequ’aussitôt , Reidemeister à utilisé cette notion pour introduire le “groupoïde fondamental (de Poincaré) d’un espace topologique” . Ehresmann a utilisé les groupoïdes de Lie dans sa construction des espaces fibrés , suivi peu de temps plus tard par Grothendieck dans sa théorie des espaces fibrés à faisceau structural . La définition usuelle d’un groupoïde comme une catégorie dans laquelle toutes les flèches sont inversibles est trop pauvre , à moins de considérer des groupoïdes enrichis dans une catégorie (notion due à Benabou) . Je décrirai les multiples développement récents , allant des feuilletages à la théorie de Galois , d’une théorie en plein développement .

Click here for link to Recording of Prof. CARTIER’s Talk

plus here for Notes of Presentation

—oOo—

10: 45 – 11:45 AM :

Professor Rene Guitart (Univ. Paris 7 Denis Diderot)

Title :  DISTRIBUTEURS ET CARRES EXACTS, AVEC APPLICATIONS A LA COHOMOLOGIE

Abstract :

La bonne gestion du calcul des fonctions a lieu au sein du calcul des relations
binaires, et le calcul de la composition des fonctions est un sous-produit de
celui des relations,
lui-même effectivement pratiqué par produits fibrés de spans représentants
lesdites relations. De même, le calcul des foncteurs est à concevoir au sien
des distributeurs et du calcul de leur composition, dans la bicatégorie ainsi
constituée. Le point clé que nous voulons mettre en relief est que la
composition effective de distributeurs se fera naturellement par produit exact
de spans, c’est-à-dire en utilisant au lieu de carrés produits fibrés des carrés exacts.

Ainsi les carrés exacts sont à comprendre comme des présentations de
compositions de distributeurs ; et, dans l’autre sens, les distributeurs
apparaissent aussi comme classe d’équivalence de spans modulos des carrés
exacts. Ensuite nous expliquerons comment les carrés exacts servent à définir
l’homologie d’une 0-suite, comment les distributeurs et les carrés exacts
servent en théorie de la forme, dans l’analyse de diverses cohomologies, pour  la définition en générale des théories de cohomologie.

Click here for link to Recording of Prof. GUITART’s Talk

plus here for Notes of Presentation

—oOo—

11:45 – 12:00 : Pause

—oOo—

12:00 – 1:00 PM  :  Professor Dominique  BOURN (Universite de Littoral, Calais)

“Protomodularité et fibration des points” 

Abstract :

Dès son article princeps sur ce qui allait devenir la notion de
catégorie abélienne (Duality forgroups Bull.AMS 1950), Mac-Lane soulignait
la nécessité d’aller au-delà du cas additif/abélien et de trouver un
axiome simple qui permettrait de conceptualiser de façon uniforme les
propriétés de la catégorie des groupes, des anneaux, des  K-algèbres etc.
Il indiquait alors l’obtention des isomorphismes de Noether comme un bon
test de réussite de ce genre de conceptualisation. C’est ce que fait
précisément la notion de catégorie protomodulaire.

Une des façons de l’introduire est de le faire via la «fibration des
points» qui se révèle avoir, par ailleurs, une remarquable puissance
classificatrice puisqu’elle permet de caractériser les catégories
additives, essentiellement affines, arithmétiques, les catégories de
Mal’cev, naturellement Mal’cev, de Gumm, protomodulaires et fortement
protomodulaires.

Click here for link to Recording of Prof. BOURN’s Talk

plus here for Notes of Presentation (Incomplete)

—oOo—

1.00 – 2: 30:PM : Lunch Interval

—oOo—

2:30 – 3:30 PM : Professor Thomas STREICHER (Darmstadt)

Title : FIBERED THEORY of GEOMETRIC MORPHISMS

Abstract :

Geometric morphisms were introduced originally by analogy with
continuous maps and locale morphisms. But as pointed out by Jean
Benabou in his Montreal lectures 1974 their definition can be
motivated in terms of “good” properties of fibrations. Later in
J.-L. Moens’ Thesis (1982) there was established a 1-1-correspondence
between geometric morphisms and so-called geometric fibrations.

After revisiting this fibered theory of geometric morphisms we
consider generalisations which allow one to formulate the theory of
triposes (Hyland et.al. around 1982) in more geometric terms. This
allows one to reformulate an open problem in the theory of triposes in
more elementary terms.

Click here for link to Recording of Prof. STREICHER’s Talk

plus here for Notes of Presentation

—oOo—

3:30 – 4:30 PM : Professor Andree C. EHRESMANN (Amiens)

Title :  DES ESPECES DE STRUCTURES LOCALES AUX DISTRUCTURES ET SYSTEMES GUIDABLES

Abstract :

Le premier article original que Jean Benabou a publié (en 1957) portait
sur les treillis locaux, reflétant son intérêt pour les travaux de Charles
Ehresmann sur les structures locales et les topologies sans points.
Je parlerai brièvement de ces travaux de Charles, et en particulier de son
important article de 1957 où il définit le cadre catégorique des espèces de
structures locales ; il y démontre le “Théorème d’élargissement complet” d’une
espèce de structures locales, dont la preuve unit une sorte d’extension de Kan à
un théorème de faisceau associé sur une “catégorie locale”.
En imitant les méthodes utilisées dans cet article, j’ai introduit (dans
ma thèse en 1962) la notion de “distructure” qui donne un cadre catégorique pour
l’étude des “fonctions généralisées”. Je l’exposerai en termes plus modernes et
montrerai comment elle permet de retrouver les distributions de Schwartz, de
manière à pouvoir définir des distributions sur des variétés de dimension
infinie. Les distructures ont des applications aux problèmes de contrôle.

Click here for link to Recording of Prof. EHRESMANN’s Talk   (Not yet uploaded. Link not yet active)

plus here for Overheads of Presentation

—oOo—

7:30 PM : Colloquium Dinner

—oOo—

SATURDAY 4th June 2011

 

9:45  – 10: 45 AM :  Professor Martin HYLAND (Univerity of Cambridge)

Title :   KLEISLI BICATEGORIES

The importance of the notion of Bicategory is widely
recognised. It has many applications and points the way
to higher category theory. Kleisli Bicategories give a setting for many sophisticated mathematical
notions. A survey of these examples illustrates the
significance of the choices made in Jean Benabou’s
Introduction to Bicategories.

Click here for link to Recording of Prof. HYLAND’s Talk   (Not yet uploaded. Link not yet active)

plus here for Notes of his Presentation and related documentation

—oOo—

10 : 45 AM : Pause

—oOo—

11: 00- 11.40 : Jacques ROUBAUD :  (Working) Title : REMEMBRANCE OF CATEGORIES’ PAST

Personal Recollections and reminiscences of Jean Benabou (En Francais)

Click here for link to Recording of  Jacques ROUBAUD’s Talk   (Not yet uploaded. Link not yet active)

—oOo—

11:45 – 1:30 PM : Lunch Interval

—oOo—

1: 30 – 2:30 PM : Professor Andre JOYAL (UQUAM)

Title :  UNIVERSAL  FIBRATIONS

Abstract:

A universal (minimal) Kan fibration was recently constructed by Voevodsky to
model Martin-Lof type theory. We shall construct a universal left fibration
and a universal Grothendieck fibration between quasi-categories.

Click here for link to Recording of Prof.JOYAL’s Talk   (Not yet uploaded. Link not yet active)

plus here for Notes of his Presentation and related documentation

—oOo—

2:30  PM :  Professor Jean BENABOU

Title :  A NEW APPROACH TO THE NOTION OF “BEING CARTESIAN”

Abstract:

Cartesian functors and maps were introduced by Grothendieck in
1960-61 for prefibrations and fibrations. Since then they have been studied
essentially only in the case of fibrations.
I shall present general definitions covering many other situations, which reduce to the classical ones in the previous cases, and give important examples which do not fit in the classical setting, and shall prove many results which were not known even in the case of fibrations.
Some of these results need assumptions, much weaker than fibrations or even
prefibrations, namely prefoliations and foliations, which I shall introduce and study briefly, with a focus on examples and counter examples.

Followed by Discussion

Click here for link to Recording of Prof. BENABOU’s Talk   (Not yet uploaded. Link not yet active)

plus here for Notes of his Presentation and related documentation

—oOo—

5:00  PM : Closure of The Colloquium.

—oOo—

Jean BENABOU : A  (very) brief Biographical Sketch and Conspectus of his Mathematical Career

Born in Rabat in Morocco in 1932, Jean Benabou is  an
ancien eleve of the ENS. He entered the School in 1952,  and his aggregation in Mathematics was in
1956. He was a CNRS Researcher from 1956 to 1962, then, after a spell at the University of Chicago
from 1966-67 at
the invitation of Saunders Mac Lane, on his return to France he directed the
Research Seminar in Category Theory (The Seminaire Benabou)  at The Institut Henri Poincare (and later
at the Institut Mathematique de Jussieu)
from 1969 to 2001. This Seminar was one of the most influential and fertile
centres in the development of Category Theory for much of that period,
attracting contributions for over 30 years from many leading researchers in the 
field.

His research career as a mathematician has now spanned over  50 years.

He commenced that career as a doctoral student of Charles Ehresmann. His
early studies in Differential Geometry and the Theory of Locales which led to his earliest published work in 1957, were motivated by a strong conceptual interest in the program of Pointless  Topology, which
was allied to his conviction as to the natural mathematical representation of Space, in particular his
rejection of the view of spaces as sets of points equipped with additional
structure. He immediately recognised the work of Grothendieck, in  particular
the Grothendieck-Giraud Theory of Topos, as providing a vast natural generalisation
of the mathematical resources for the representation of space, going  beyond
the localic case.
 
These interests led to the publication of 3 Memoirs in The Comptes Rendues
(1963-65) prior to the completion of his Thesis ( Title : Structures 
Algebriques dans les Categories 1966 ). These Notes and the Thesis introduced the notions of Enriched 
Categories, Monoidal Categories and Monoidal Functors – the latter being the earliest examples of
lax functors, the study of which has since become a central focus of research in Category Theory, not least in
connection with the applications of Category Theory to Theoretical Computer
Science.

As mentioned above, an invitation from Saunders Mac Lane took him to the University of Chicago,
where in 1966-67 he set out to generalise his earlier work on Monoidal and
Enriched Categories via the Theory of Bicategories, as the setting  for the
study of Distributeurs. These contributions signposted the way towards higher
category theory,  the topic of a large portion of current research in the
discipline, and regarded by many today as its main axis of development

His work on the Theory of Distributeurs, including Enriched and Internal
Distributeurs, led to an increasing engagement with Descent Theory. He published a notable memoir surveying this field together wih Jacques Roubaud in 1970.  This engagement helped give rise
to his extensive work in Fibred Categories which was the primary focus
of his research in the 1970s and 80s. His Course on Fibred
Categories given at Louvain-La-Neuve in 1980  and his 1985 JSL Paper  on
Fibred Categories are regarded as landmarks in the development of that
subject while his work during the same period in Topos Theory and on other
aspects of the applications of Category Theory to Logic reflected
his view of Topos Theory as essentially a chapter within the Theory  of
Fibered Categories. His Report of 1971 with Jean Ceylerette was an early systematic
exposition of the then – new Lawvere-Tierney Topos Theory, whilst
amongst his contributions to Categorial Logic was his work on the clarification of the
Internal Language of a Topos, and the Categorial study of Syntax,  including  the
Theory of Sketches, and latterly of Generalised Sketches.

The last mentioned subject is one aspect of the Theory of Foliated
Categories – a far-reaching generalisation of Fibred Categories setting out from the
notions of Cartesian Functors and Homotopy pre-fibrations, which he has been
pursuing in recent years and which is today the principal focus of  his
research.
 
He has also worked in the Theory of Allegories and has pursued  research in
areas of mathematics outside Category Theory, including Non-Standard Analysis.  Throughout his career he has also nurtured a close interest in general foundational questions, particularly the
respective foundational roles of categorial constructions and set theory, on
which his work on Fibred and Foliated Categories has an important bearing.
 
Already in his sixth decade of research, his appetite and enthusiasm for
mathematics is unwavering and the energy powering his search – in particular for the simplest minimal assumptions underlying the most powerful, unifying notions of mathematics – a search strongly guided by his
sense of mathematical beauty and elegance – is entirely undimmed