Category theory and its applications

A conference in memory of Saunders Mac Lane

7-11 April, 2006
University of Chicago



The conference will be based at University of Chicago, with talks being held at the Department of Mathematics building which is called Eckhart.


Information for participants

Click here for local and campus maps (pdf).

The programme will begin at 9am on Friday April 7, with coffee and donuts in the Eckhart (math building) Tea Room.  The easiest way to find the Tea Room is to enter Eckhart by the entrance marked on the map, go up the stairs to the second floor and turn left.  You'll see the Tea Room almost immediately on the left.

The Memorial Service will be held at 10am in the Bond Chapel, just across the Quadrangle.  This will be followed by a buffet lunch reception at the Quadrangle Club on 57th Street.




Contacts

Any queries may be addressed to the organisers:

Eugenia Cheng (eugenia at math.uchicago.edu)
Peter May (may at math.uchicago.edu)

Logistical questions should be addressed to Eugenia Cheng.
For urgent contact during your trip to Chicago, you can call Eugenia's cell phone on 1-312-730-3445.  



On this website you also can find the following information:


And below:

Schedule
Namboodiri Lectures: abstract
Participants



Schedule
(revised)


Friday 7 Saturday 8
Sunday 9
Monday 10
Tuesday 11
10:00--12:00
MacLane
Memorial
9:30--10:30
Johnstone
11:00--12:00
Lawvere
9:30--10:30
Stevenson
11:00--12:00
Fiedorowicz
9:30--10:30
Gurski
11:00--12:00
Bergner
9:30--10:30
Fiore
11:00--12:00
Shulman
2:00--3:30
Reminiscences
2:00--3:00
Crans
2:00--3:00
Moerdijk

2:00--3:00
Awodey

1:30--2:30
Freyd
3:00--4:00
May
4:00--5:00
Baez/Namb.
4:00--5:00
Joyal
4:00--5:00
Cheng
4:00--5:00
Baez/Namb.
4:30--5:30
Baez/Namb.


Details
morning and afternoon coffee breaks will take place in the Eckhart Tea Room

Friday April 7



9:00--10:00
Coffee and donuts

Eckhart Tea Room
10:00--12:00
Mac Lane Memorial

Bond Chapel

Reception

Quadrangle Club
2:00--3:30
Reminiscences

Ry 352 (Barn)

Coffee

Eckhart Tea Room
4:00--5:00
John Baez, Namboodiri I
Higher categories, higher gauge theory
Ry 251
Saturday April 8



9:30--10:30
11:00--12:00
Peter Johnstone
William Lawvere
Potential invertibility and presheaf toposes
Smooth and simplicial toposes
Ry 251
2:00--3:00
4:00--5:00
Alissa Crans
André Joyal
Lie 2-groups, Lie 2-algebras, and Loop groups
The theory of quasi-categories
Ry 251
Sunday April 9



9:30--10:30
11:00--12:00
Danny Stevenson
Zig Fiedorowicz
Lie 2-algebras and the geometry of gerbes
Tensor products of E_n operads
Ry 251
2:00--3:00
4:00--5:00
Ieke Moerdijk
Eugenia Cheng
Quasi-categories and quasi-operads
The periodic table of n-categories
Ry 251
Monday April 10



9:30--10:30
11:00--12:00
Tom Fiore
Julie Bergner
Double categories and pseudo algebras
Model categories, dg categories, and derived Hall algebras
Ry 352 (Barn)
2:00--3:00
4:00--5:00
Steve Awodey
John Baez, Namboodiri II
Topology and modality
Higher categories, higher gauge theory
Ry 352 (Barn)
Ry 251
Tuesday April 11



9:30--10:30
11:00--12:00
Nick Gurski
Mike Shulman
From bicategories to tricategories
Anchored bicategories
Ry 352 (Barn)
1:30--2:30
3:00--4:00
4:30--5:30
Peter Freyd
Peter May
John Baez, Namboodiri III
New structures on old categories
Duality in bicategories and topological applications
Higher categories, higher gauge theory
Eck 206





Namboodiri Lectures
Higher categories, higher gauge theory
John Baez
University of California at Riverside

Notes are available here.

Abstract:

The work of Eilenberg and Mac Lane marks the beginning of a trend in which mathematics based on sets is generalized to mathematics based on categories and then higher categories.  We illustrate this trend towards "categorification" by a detailed introduction to "higher gauge theory".

Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles.  In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space.  This suggests that we seek some kind of "higher gauge theory" that describes the parallel transport as we move a path through space, tracing out a surface.  Surprisingly, this requires that we "categorify" concepts from differential geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and so on.  The basic tool used here is Ehresmann's notion of "internalization".

To explain how higher gauge theory fits into mathematics as a whole, we begin with a lecture reviewing the basic principle of Galois theory and its relation to Klein's Erlangen program, covering spaces and the fundamental group, Eilenberg-Mac Lane spaces, and Grothendieck's ideas on fibrations.

The second lecture treats connections on trivial bundles and 2-connections on trivial 2-bundles, explaining how they can be described either in terms of their holonomies or in terms of Lie-algebra-valued differential forms. For a clean treatment of these concepts, we recall Chen's theory of "smooth spaces", which generalize smooth finite-dimensional manifolds.

The third lecture explains connections on general bundles and 2-connections on general 2-bundles, explaining how they can be described either in terms of holonomies or local data involving differential forms.  We also explain how 2-bundles are classified using nonabelian Cech 2-cocycles, and how the theory of 2-connections relates to Breen and Messing's theory of "connections on nonabelian gerbes".



Participants

to be updated regularly


Name
Affiliation        
Steve Awodey
Carnegie Mellon University
John Baez
University of California at Riverside
Julie Bergner
Kansas State University
Jeff Caruso

Eugenia Cheng
University of Chicago
Alissa Crans
Ohio State University
Geoff Cruttwell
Dalhousie University
Robert Dawson
St Mary's University
Thomas Drucker
University of Wisconsin--Whitewater
Tony Elmendorf
Purdue University at Calumet
Zbigniew Fiedorowicz
Ohio State University
Thomas Fiore
University of Chicago
Brandon Fogel
University of Notre Dame
Peter Freyd
University of Pennsylvania
Carl Futia

Megan Guichard
University of Chicago
Bertrand Guillou
University of Chicago
Nick Gurski
University of Chicago
Michele Intermont
Kalamazoo College
Samuel Isaacson
Harvard University
Rick Jardine
University of Western Ontario
Niles Johnson
University of Chicago
Peter Johnstone
Cambridge University
André Joyal
Université du Québec à Montréal
Noah Kieserman
University of Wisconsin--Madison
Joachim Kock
Universitat Autònoma de Barcelona
Sanjeevi Krishnan
University of Chicago
Aaron Lauda
University of Cambridge
William Lawvere
SUNY at Buffalo
Dean Leonardi
University of Illinois at Chicago
Michael Lieberman
University of Michigan
John Macdonald
University of British Columbia
Michael Mandell
Indiana University
Howard Marcum
The Ohio State University at Newark
Martin Markl
Academy of Sciences of the Czech Republic
Peter May
University of Chicago
Ieke Moerdijk
University of Utrecht
Justin Noel
University of Chicago
Son Nguyen
Wayne State University
Zaza Omiadze
A.Razmadze Mathematical Institute, Tbilisi
Bob Paré
Dalhousie University
Hendryk Pfeiffer
Max Planck Institute for Gravitational Physics
Kate Ponto
University of Chicago
Dorette Pronk
Dalhousie University
Laura Scull
University of British Columbia
Brooke Shipley
University of Illinois at Chicago
Armira Shkembi
Wayne State University
Michael Shulman
University of Chicago
Jim Stasheff
University of Pennsylvania
Danny Stevenson
University of Adelaide
Lawrence Stout
Illinois Wesleyan University





This site is maintained by Eugenia Cheng.  You can e-mail me at: eugenia at math.uchicago.edu

Last updated April 6 2006, 21:31