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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 2category "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 openended discussions. As a general rule, we are following an old suggestion to have particular definitions of (weak) ncategories 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.




MONDAY,
JUNE 7 All talks are in Lecture Hall EE/CS 3180 unless otherwise noted. 


8:309:00  Coffee and Registration  Reception Room EE/CS 3176 

9:009:15  Douglas N. Arnold, Scot Adams, and Organizers  Welcome and Introduction  
9:1510:45  John
Baez University of California at Riverside 

10:4511:15  Coffee Break  Reception Room EE/CS 3176  
11:1512:15  John
Baez University of California at Riverside 

2:00210  
2:10  Reception  400 Lind Hall  
4:008:00  Thomas
Leinster University of Glasgow 
Survey and Taxonomy  
TUESDAY,
JUNE 8 *** All talks are in "Smith Hall 331" unless otherwise noted. 

8:309:00  Coffee 


9:0010:30  André
Joyal Université du Québec à Montréal J. Peter May University of Chicago Timothy Porter University of Wales, Bangor 
Weak Categories 

10:3011:00  Coffee Break  
11:0012:00  Michael
Batanin Macquarie University 
A_{ }Operad Definition  
4:008:00  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. 

8:309:00  Coffee 


9:0010:30  J.
Peter May University of Chicago 
Model Categories 

10:3011:00  Coffee Break  
11:0012:00  Clemens
Berger University of Nice 
Cellular Definitions Slides: pdf 

4:008:00  Eugenia
Cheng University of Cambridge 
Multicategories and Related Definitions and Opetopic Definitions  
THURSDAY,
JUNE
10 All talks are in Lecture Hall EE/CS 3180 unless otherwise noted. 

8:309:00  Coffee  Reception Room EE/CS 3176 

9:0010:30  André
Joyal Université du Québec à Montréal 
Perspective  
10:3011:00  Coffee Break  Reception Room EE/CS 3176  
11:0012:00  Nick
Gurski University of Chicago Thomas Leinster University of Glasgow 
Slides: streettalk1.pdf streettalk2.pdf 

4:008:00  Larry
Breen Universite Paris 13 
nstacks& ngerbes: Homotopy Theory 

FRIDAY,
JUNE 11 All talks are in Lecture Hall EE/CS 3180 unless otherwise noted. 

8:309:00  Coffee  Reception Room EE/CS 3176 

9:0010:30  David
Neil Corfield Oxford University 
NCategory Theory as a Catalyst for Change in Philosophy 

10:3011:00  Coffee Break  Reception Room EE/CS 3176  
11:0012:00  Bertrand
Toën Universite Paul Sabatier 
Segalic Definition Notes taken by Joachim Kock pdf 

4:006:00  Bertrand
Toën Universite Paul Sabatier 
nstacks& ngerbes: Algebraic Geometry Notes taken by Joachim Kock pdf 

MONDAY,
JUNE 14 All talks are in Lecture Hall EE/CS 3180 unless otherwise noted. 

8:309:00  Coffee  Reception Room EE/CS 3176 

9:0010:30  Zbigniew
Fiedorowicz Ohio State University 
nFold Categories Slides: pdf 

10:3011:00  Coffee Break  Reception Room EE/CS 3176  
11:0012:00  Stefan
Forcey Virginia Tech 
Higher Enrichment: NFold Operads and Enriched NCategories, Delooping and Weakening 

4:008:00  moderator 
Comparisons  
TUESDAY,
JUNE 15 All talks are in Lecture Hall EE/CS 3180 unless otherwise noted. 

8:309:00  Coffee  Reception Room EE/CS 3176 

9:0010:30  Michael
Makkai McGill University 
Perspectives


10:3011:00  Coffee Break  Reception Room EE/CS 3176  
11:0012:00  Mark
Weber University of Ottawa 
Higher Monads  
4:008:00  moderator 
Comparisons  
WEDNESDAY,
JUNE 16 All talks are in Lecture Hall EE/CS 3180 unless otherwise noted. 

8:309:00  Coffee  Reception Room EE/CS 3176 

9:0010:30  Michael
Batanin Macquarie University 
Perspectives  
10:3011:00  Coffee Break  Reception Room EE/CS 3176  
11:0012:00  Joachim
Kock Université du Québec à Montréal 
Topological Quantum Field Theory Primer Summary and Slides: pdf 

4:008:00  Topological
Quantum Field Theory and Quantum Gravity


THURSDAY,
JUNE 17 All talks are in Lecture Hall EE/CS 3180 unless otherwise noted. 

8:309:00  Coffee  Reception Room EE/CS 3176 

9:0010:30  Ross
Street Macquarie University 
An Australian Conspectus of Higher Categories Notes: pdf 

10:3011:00  Coffee Break  Reception Room EE/CS 3176  
11:0012:00  Stephen
Lack University of Western Sydney 
Higher Model Categories Notes:
pdf


4:008:00  Computer
Science: Programming Languages and Concurrency


FRIDAY,
JUNE 18 All talks are in Lecture Hall EE/CS 3180 unless otherwise noted. 

8:309:00  Coffee  Reception Room EE/CS 3176 

9:0010:30  Martin
Hyland Cambridge University 
Perspectives  
10:3011:00  Coffee Break  Reception Room EE/CS 3176  
11:0012:00  Alissa
Crans University of California at Riverside 
Higher Linear Algebra Slides: pdf 

2:004:00  Tutti Problems and Directions 
Week
1 6/76/11 
Monday  Tuesday  
Week
2 6/146/18 
Monday  Tuesday 
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 DenisDiderot  
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 
HyeungJoon 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 
JeanPierre Meyer  Department of Mathematics  The Johns Hopkins University 
Gary Nan Tie  The St. Paul Companies  
Joshua Paul NicholsBarrer  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 