The Morgan-Phoa Mathematics Workshop
Dates: Tuesday 28, Wednesday 29 and Thursday 30 November 2006
Venue: Australian National University, Canberra: Room JD 101 of the John Dedman Building is booked 9am--5pm Tuesday through Thursday.
Purpose: It has been clear for several years that CoACT (Macquarie University) and Amnon Neeman's group at the CMA (Australian National University) have many common research interests. The plan for this Workshop was to investigate these connections, and to advance those research areas, in an informal and flexible setting. Some of these common interests include categorical homotopy theory, topos theory, triangulated categories, K-theory, higher categories, homological algebra, cohomology, and differential graded categories.
Participants included: Dorette Pronk, Marni Sheppeard, Amnon Neeman, Michael Batanin, Stephen Lack, James Borger, Alexei Davydov, Ross Street, Dimitri Chikhladze, Craig Pastro, Daniel Murfet, Daniel Steffen, Jon Cohen, Steve Bennoun, Thorsten Palm, Carolyn Kennett, Mark Weber, Boris Chorny, Simona Paoli, Peter Bouwknegt. Many more attended the Colloquium.
Formal talks:
Ross Street spoke on Tuesday 4-5pm in the ANU Algebra Seminar (Room G35) . The topic was Quantum categories, Frobenius algebras and weak Hopf algebras in braided monoidal categories based on joint work with Craig Pastro. Abstract below.
Michael Batanin spoke on Thursday 4-5pm in the ANU Mathematics Colloquium (Room G35) . The topic was Deligne's conjecture: an interplay between algebra, geometry and higher category theory . Abstract below. Available as pdf .
Report on the Workshop
We met each morning at 9:30 am. There was a break for coffee, tea, and lovely cake at 11 am. We broke for lunch between (roughly) 1:15 to 2 pm and continued until more coffee, tea, and lovely cake at 3 p m. On Tuesday and Thursday at 4 pm were the formal talks. On Wednesday we resumed the Workshop after the afternoon tea break until 5:30 pm. The Algebra Seminar Dinner began at 6 pm Tuesday while the Colloquium Dinner began 6 pm Wednesday (well before the Colloquium!) so that those who needed to could leave Thursday evening.
Presentations included the following:
Tuesday
Stephen Lack: Introduction to topos theory.
James Borger: Algebraic spaces, what are they categorically?
Boris Chorney: What kind of "topos" is the category of small presheaves on a locally small category (such as the category of orbits of a small category)?
Mark Weber: Introduction to 2-toposes.
Wednesday
Marni Sheppeard: Operads and polylogarithms in physics.
Michael Batanin: Modelling of homotopy types.
Simona Paoli: A model structure on internal categories and applications to homotopy types.
Alexei Davydov: Autoequivalences of categories of representations of groups.
Amnon Neeman: Is there a general theorem about when two derived categories are equivalent as triangulated categories? Examples are known in representation theory and in algebraic geometry but the proofs are particular.
Michael Batanin: Gray operads.
Thursday
Peter Bouwknegt: What is the natural definition of a C*-algebra in a monoidal category (with suitable structure)? The T-dual of a C*-algebra should be such.
Alexei Davydov: To what extent is the K of K-theory monoidal?
Boris Chorny: The Goodwillie Calculus of Functors.
James Borger: Witt vectors and lambda-rings.
Accommodation: Most people stayed at Toad Hall and some at Argyle Apartments . The John Dedman Building is less than a 300 meter walk from Toad Hall and less than 20 minutes walk from the Argyle Apartments.
Support: Margaret Morgan and Wesley Phoa
Centre for Mathematics and its Applications
Centre of Australian Category Theory
Mathematics Department, Macquarie University
Title: Quantum categories, Frobenius algebras and weak Hopf algebras in braided monoidal categories
Speaker: Ross Street (joint work with Craig Pastro)
Abstract:
The monoid algebra kM of a monoid M over the field k has a comultiplication (coming from the diagonal M --> M x M ); the resultant structure on kM is that of a bialgebra. The comultiplication of a bialgebra A can be used to obtain a monoidal structure on the category Rep A of linear representations of the algebra A . If M = G is a group then kG has an antipode (coming from taking inverses); the resultant structure is that of Hopf algebra and Rep H behaves very much like the category of vector spaces (especially if H has an integral). If G is a finite group then kG is also a Frobenius algebra. The main examples of quantum groups are Hopf algebras H for which Rep H is braided. Motivated by the fact that an algebra is a monoid in the category of vector spaces, we can now bootstrap and look at a monoid A in Rep H and the category of representations of A in Rep H ; this last is a category of linear representations of the semidirect product of A and H . It therefore is worth abstracting all these concepts to braided monoidal categories (BMCs).
Several directions of research have led people to require groupoids as well as the one-object case of groups. By now there is a serious theory of quantum groupoids with many examples. We look at the concepts of quantum category, quantum groupoid, weak bimonoid, Frobenius monoid, and weak Hopf monoid in a general BMC. We study various relationships between the concepts. We make extensive use of the rigorous theory of string diagrams in BMCs.
Title: Deligne's conjecture: an interplay between algebra, geometry and higher category theory
Speaker: Michael Batanin
Abstract:
In 1993 Deligne made a conjecture that the Hochschild complex of an associative algebra (a purely algebraic object) has a natural action of the chain of the little disks operad (a purely geometric object). The first correct proof due to Tamarkin appeared in 1998. After that, many other proofs were found but the story of Deligne's conjecture is still far from over.
In my lecture I will explain the conjecture, its importance in non-commutative differential calculus and why it is so inspiring. I also will give a brief sketch of one of the most recent proofs based on a combination of the results of Tamarkin and Batanin.