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IMA 2004 Summer Program:
n-Categories: Foundations and Applications
June 7-18, 2004


Organizers:

John C. Baez
Department of Mathematics
University of California at Riverside
baez@math.ucr.edu
http://www.math.ucr.edu/home/baez/

J. Peter May
Department of Mathematics
University of Chicago
may@math.uchicago.edu
http://www.math.uchicago.edu/~may/

This summer program is about an area of mathematics which is very new and of great current interest. It concerns iterative structures that appear naturally in a wide variety of contexts.

Eilenberg and MacLane introduced categories, functors, and natural transformations in a landmark 1945 paper. The language they introduced transformed modern mathematics. In fact, a very great deal of mathematics since then would quite literally have been unthinkable without that language. Their focus was not on categories and functors, but on natural transformations, which are maps between functors. Implicitly, they were introducing the 2-category "Cat" of categories, functors, and natural transformations.

Higher category theory concerns higher level notions of naturality, which can be expressed as maps between natural transformations, maps between such maps. Just as the original definitions of Eilenberg and MacLane gave a way of thinking about categorical structures and analogies between such structures in different fields, higher category theory promises to allow serious thinking about and study of higher categorical structures that appear in a variety of specific fields. The need for such a language has become apparent, almost simultaneously, in mathematical physics, algebraic geometry, computer science, logic, and, of course, category theory. Such a language and a relevant body of results is already implicit throughout algebraic topology. In all of these areas, higher categorical structures are there in nature, and one needs a coherent way of thinking about them.

In contrast to the introduction of categories, functors, and natural transformations, which could successfully be carried out by two authors in one paper, the development of higher category theory is technically very difficult mathematics. Our central goal is to end up with a coherent theory that is field independent. We hope to develop a clear language of higher category theory that, like the original language of categories, functors, and natural transformations, can be accepted, understood, and worked with by mathematicians in general, whether algebraic geometers, logicians, algebraic topologists, mathematical physicists, or theoretical computer scientists.

We have in mind many general surveys and open-ended discussions. As a general rule, we are following an old suggestion to have particular definitions of (weak) n-categories presented by people other than their creators. Following the lead of experience with the geometric Langlands seminar at Chicago, and other "Russian style'' seminars elsewhere, we plan long late afternoon and evening sessions. We are in process of working out logistics for dinners at these seminars. (Going out for dinner is civilized, but fattening and not very productive.) Early afternoons are working time.

SUMMER WORKSHOP SCHEDULE
Week 1
6/7-6/11
Monday Tuesday
Week 2
6/14-6/18
Monday Tuesday
First Week
MONDAY, JUNE 7
All talks are in Lecture Hall EE/CS 3-180 unless otherwise noted.
8:30-9:00 Coffee and Registration

Reception Room EE/CS 3-176

9:00-9:15 Douglas N. Arnold, Scot Adams, and Organizers Welcome and Introduction
9:15-10:45 John Baez
University of California at Riverside

Why n-Categories?

Slides:   pdf    ps

10:45-11:15 Coffee Break Reception Room EE/CS 3-176
11:15-12:15 John Baez
University of California at Riverside

What n-Categories Should Be Like

Slides:   pdf    ps

2:00-210
Group Photos here
2:10 Reception 400 Lind Hall
4:00-8:00
Thomas Leinster
University of Glasgow
Survey and Taxonomy
TUESDAY, JUNE 8
*** All talks are in "Smith Hall 331" unless otherwise noted.
8:30-9:00 Coffee

 

9:00-10:30 André Joyal
Université du Québec à Montréal
J. Peter May
University of Chicago
Timothy Porter
University of Wales, Bangor

Weak Categories

Slides:   pdf
Timothy Porter Notes:
  pdf

10:30-11:00 Coffee Break  
11:00-12:00 Michael Batanin
Macquarie University
Ainfinity Operad Definition
4:00-8:00 Thomas Leinster
University of Glasgow
Eugenia Cheng
University of Cambridge
Multicategories and Related Definitions
WEDNESDAY, JUNE 9
*** All talks are in "Smith Hall 331" unless otherwise noted.
8:30-9:00 Coffee

 

9:00-10:30 J. Peter May
University of Chicago

Model Categories

Slides:   pdf    ps

10:30-11:00 Coffee Break  
11:00-12:00 Clemens Berger
University of Nice

Cellular Definitions

Slides:   pdf

4:00-8:00 Eugenia Cheng
University of Cambridge
Multicategories and Related Definitions and Opetopic Definitions
THURSDAY, JUNE 10
All talks are in Lecture Hall EE/CS 3-180 unless otherwise noted.
8:30-9:00 Coffee

Reception Room EE/CS 3-176

9:00-10:30
André Joyal
Université du Québec à Montréal
Perspective
10:30-11:00 Coffee Break Reception Room EE/CS 3-176
11:00-12:00 Nick Gurski
University of Chicago
Thomas Leinster
University of Glasgow

Simplicial Definition

Slides: streettalk1.pdf    streettalk2.pdf

4:00-8:00 Larry Breen
Universite Paris 13

n-stacks& n-gerbes: Homotopy Theory

FRIDAY, JUNE 11
All talks are in Lecture Hall EE/CS 3-180 unless otherwise noted.
8:30-9:00 Coffee

Reception Room EE/CS 3-176

9:00-10:30 David Neil Corfield
Oxford University

N-Category Theory as a Catalyst for Change in Philosophy

Notes:   pdf
Slides:   pdf    doc

10:30-11:00 Coffee Break Reception Room EE/CS 3-176
11:00-12:00 Bertrand Toën
Universite Paul Sabatier

Segalic Definition

Notes taken by Joachim Kock   pdf

4:00-6:00 Bertrand Toën
Universite Paul Sabatier

n-stacks& n-gerbes: Algebraic Geometry

Notes taken by Joachim Kock   pdf

Second Week
MONDAY, JUNE 14
All talks are in Lecture Hall EE/CS 3-180 unless otherwise noted.
8:30-9:00 Coffee

Reception Room EE/CS 3-176

9:00-10:30 Zbigniew Fiedorowicz
Ohio State University

n-Fold Categories

Slides:   pdf

10:30-11:00 Coffee Break Reception Room EE/CS 3-176
11:00-12:00 Stefan Forcey
Virginia Tech

Higher Enrichment: N-Fold Operads and Enriched N-Categories, Delooping and Weakening

Slides:   pdf
Paper:
  pdf

4:00-8:00
moderator
Comparisons
TUESDAY, JUNE 15
All talks are in Lecture Hall EE/CS 3-180 unless otherwise noted.
8:30-9:00 Coffee

Reception Room EE/CS 3-176

9:00-10:30 Michael Makkai
McGill University

Perspectives

 

10:30-11:00 Coffee Break Reception Room EE/CS 3-176
11:00-12:00 Mark Weber
University of Ottawa
Higher Monads
4:00-8:00
moderator
Comparisons
WEDNESDAY, JUNE 16
All talks are in Lecture Hall EE/CS 3-180 unless otherwise noted.
8:30-9:00 Coffee

Reception Room EE/CS 3-176

9:00-10:30 Michael Batanin
Macquarie University
Perspectives
10:30-11:00 Coffee Break Reception Room EE/CS 3-176
11:00-12:00 Joachim Kock
Université du Québec à Montréal

Topological Quantum Field Theory Primer

Summary and Slides:   pdf

4:00-8:00
Topological Quantum Field Theory and Quantum Gravity
Marco Mackaay
University do Algarve
Slides:   pdf1    pdf2
John Baez
University of California at Riverside

Space and State, Spacetime and Process

Slides:   pdf    ps

THURSDAY, JUNE 17
All talks are in Lecture Hall EE/CS 3-180 unless otherwise noted.
8:30-9:00 Coffee

Reception Room EE/CS 3-176

9:00-10:30
Ross Street
Macquarie University

An Australian Conspectus of Higher Categories

Notes:   pdf

10:30-11:00 Coffee Break Reception Room EE/CS 3-176
11:00-12:00 Stephen Lack
University of Western Sydney

Higher Model Categories

Notes:   pdf
Papers:

A Quillen Model Structure for 2-Categories

A Quillen Model Structure for Bicategories

4:00-8:00
Computer Science: Programming Languages and Concurrency
John Power
University of Edinburgh

Why Tricategories?

Paper:   pdf

Philippe Gaucher
Université Paris 7 Denis-Diderot

Towards a Homotopy Theory of Higher Dimensional Automata

Slides:   pdf

Lisbeth Fajstrup
Aalborg University
Colored Slides:
Ovh1.pdf
Ovh2.pdf
Ovh3.pdf
Ovh4.pdf
Ovh5.pdf
Ovh6.pdf
Black & White Slides:   pdf
Short note explaining the slides:   pdf
Eric Goubault
Commissariat l'Energie Atomique
Slides:  pdf   ps
 
FRIDAY, JUNE 18
All talks are in Lecture Hall EE/CS 3-180 unless otherwise noted.
8:30-9:00 Coffee

Reception Room EE/CS 3-176

9:00-10:30 Martin Hyland
Cambridge University
Perspectives
10:30-11:00 Coffee Break Reception Room EE/CS 3-176
11:00-12:00 Alissa Crans
University of California at Riverside

Higher Linear Algebra

Slides:   pdf

2:00-4:00 Tutti    Problems and Directions  
Week 1
6/7-6/11
Monday Tuesday
Week 2
6/14-6/18
Monday Tuesday

LIST OF CONFIRMED PARTICIPANTS

Name Department Affiliation
Scot Adams Institute for Mathematics and its Applications University of Minnesota
Douglas N. Arnold Institute for Mathematics and its Applications University of Minnesota
Donald G. Aronson Institute for Mathematics and its Applications University of Minnesota
Steve Awodey Department of Philosophy Carnegie Mellon University
Nils A. Baas Department of Mathematical Sciences Norewegian University of Science and Technology
Bernard Badzioch School of Mathematics University of Minnesota
John Baez Department of Mathematics University of California - Riverside
Antar Bandyopadhyay   University of Minnesota
Michael Batanin Department of Mathematics Macquarie University
Clemens Berger Laboratoire J.A. Dieudonné University of Nice
Julie Bergner Department of Mathematics Notre Dame University
William Boshuck Department of Mathematics John Abbott College
Larry Breen "Laboratoire Analyse, Géométrie et Applications" Universite Paris 13
Ronnie Brown Department of Mathematics University of Wales
Manuel Bullejos Departamento de Álgebra University of Granada
Jeffrey L. Caruso   Pageflex, Inc.
Sunil Chebolu Department of Mathematics University of Washington
Eugenia Cheng Department of Pure Mathematics and Mathematical Statistics Cambridge University
David Neil Corfield Department of Philosophy Oxford University
Alissa Crans Department of Mathematics University of California - Riverside
Alexei Davydov Department of Mathematics Macquarie University
Aurora Ines Del Rio Cabeza Algebra Department University of Granada
James Dolan Department of Mathematics University of California - Riverside
Josep Elgueta Departament Matematica Aplicada II Universitat Politecnica de Catalunya
Anthony D. Elmendorf Department of Mathematics Purdue University
Ulrich Fahrenberg Department of Mathematical Sciences Aalborg University
Lisbeth Fajstrup Department of Mathematical Sciences Aalborg University
Mark Feshbach Department of Mathematics University of Minnesota
Zbigniew Fiedorowicz Department of Mathematics Ohio State University
Thomas M. Fiore Department of Mathematics University of Michigan
Stefan Forcey Department of Mathematics Virginia Tech
Carl A. Futia    
Philippe Gaucher   Université Paris 7 Denis-Diderot
Eric Goubault   Commissariat l'Energie Atomique
Nick Gurski Department of Mathematics University of Chicago
Eric Harrelson School of Mathematics University of Minnesota
Mehdi Hakim Hashemi School of Mathematics University of Minnesota
Claudio Hermida School of Computing Queen's University
Thomas Hunter Department of Mathematics and Statistics Swarthmore College
Martin Hyland Department of Pure Mathematics and Mathematical Statistics Cambridge University
Michael Johnson Department of Mathematics and Computer Science Macquarie University
André Joyal Shannon Laboratory Département de mathématiques Université du Québec à Montréal
Hyeung-Joon Kim School of Mathematics University of Minnesota
Joachim Kock Department of Mathematics Universite du Quebec a Montreal
Wojciech Komornicki Department of Mathematics Hamline University
Sanjeevi Krishnan Department of Mathematics University of Chicago
Stephen Lack School of Quantitative Methods and Mathematical Sciences University of Western Sydney
Yves Lafont Institut de Mathématiques de Luminy Université de la Méditerranée (Marseille)
Aaron Lauda Department of Mathematics University of California - Riverside
Tom Leinster Department of Mathematics University of Glasgow
Marco Mackaay Area Departamental de Matematica University do Algarve
Michael Makkai Department of Mathematics McGill University
Gianfranco Mascari   Consiglio Nazionale delle Ricerche (CNR)
J. Peter May Department of Mathematics University of Chicago
William Messing School of Mathematics University of Minnesota
Francois Metayer Department of Mathematics Equipe PPS Universite Paris 7
Jean-Pierre Meyer Department of Mathematics The Johns Hopkins University
Gary Nan Tie   The St. Paul Companies
Joshua Paul Nichols-Barrer Department of Mathematics Massachusetts Institute of Technology
Simona Paoli Mathematics Institute University of Warwick
Timothy Porter School of Informatics University of Wales
John Power School of Informatics University of Edinburgh
Victor Reiner Department of Mathematics University of Minnesota
Tony Robbin   Yale University Press
Jonathan Rogness School of Mathematics University of Minnesota
Andrew Michael Salch Department of Mathematics University of Rochester
Fadil Santosa Institute for Mathematics and its Applications University of Minnesota
Dana Stewart Scott Department of Computer Sciences Carnegie Mellon University
Michael Shulman Department of Mathematics University of Chicago
Richard Steiner Department of Mathematics University of Glasgow
Danny Stevenson Department of Pure Mathematics University of Adelaide
Ross Street Department of Mathematics Macquarie University
James Swenson School of Mathematics University of Minnesota
Bertrand Toen Laboratoire J.A. Dieudonne University of Nice
Gabriele Vezzosi Dipartimento di Matematica Applicata 'G. Sansone' Università degli Studi di Firenze
Alexander A. (Sasha) Voronov School of Mathematics University of Minnesota
Peter Webb School of Mathematics University of Minnesota
Mark Weber Department of Mathematics and Statistics University of Ottawa
Ittay Weiss Department of Mathematics Utrecht University
Marek Zawadowski Department of Mathematics University of Warsaw
Javier Zuniga   University of Minnesota
Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities, September 1, 2004 - June 30, 2005
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