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 Wednesday Thursday Friday Week 2 6/14-6/18 Monday Tuesday Wednesday Thursday Friday
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	A 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

LIST OF CONFIRMED PARTICIPANTS