Projects

Projects undertaken in 2003

Category theory arising from geometry, algebra, computer science and physics

Personnel: Ross Street (Chief Investigator), Max Kelly (CI), Michael Johnson (CI), Stephen Lack (CI), Brian Day, Michael Batanin, George Janelidze, Isar Stubbe, Zurab Janelidze, John Corbett., Daniel Steffen, Catherine Menon.

Summary: Category theory is a branch of mathematics concerned with transformation and composition. It provides an algebra of wide-spread applicability for the synthesis of systems and processes in fields as diverse as geometry, physics and computer science, and also in mathematics itself. Often it can be used to clarify and simplify the learning, teaching and development of mathematics. The aim of this project is to develop the general theory of categories and specifically to investigate aspects appropriate to algebra, physics and computer science.

Invariants of higher-dimensional categories, with applications

Personnel: Ross Street (CI), Alexei Davydov, John Baez, Simona Paoli.

Summary: Complex systems in mathematics are difficult to tell apart so one constructs simpler structures from them. These structures must be equal, isomorphic or equivalent when the original systems are equivalent; the word invariant is used for such constructions. Higher-dimensional categories are complex structures that are currently gaining a lot of attention from mathematicians, physicists and computer scientists because of developing applications in those fields. This project will establish and study invariants for higher-dimensional categories which will be tested by examining their viability for producing results in group theory and homotopy theory.

Categorical universal algebra and the foundations of information management

Personnel: Michael Johnson (CI), Kit Dampney (in 2003), Kate Krastev (in 2003), Jason Rennie, Catherine Menon (in 2004).

Summary: Information management depends upon providing structures on data stored in databases, semi-structured documents, formal specifications and libraries. Different structured arrangements of equivalent data frequently lead to apparent incompatibilities and limit the interoperations possible between systems. Categorical universal algebra has confronted this problem of inequivalent specifications of equivalent data in the context of algebraic structures. In preliminary work we have shown how analogous categorical techniques for the mathematical specification and analysis of structured data can be used to obtain data invariants which aid in the development of system interoperations. This project develops those techniques and extends them to semi-structured data and formal specifications.

Operadic techniques in the theory of crossed modules and hypercrossed complexes.

Personnel: Stephen Lack (CI), Simona Paoli.

Summary: Geometry concerns the spatial relationships between objects in the real world. Algebra involves calculations and formal manipulations of symbols; it has been described as a Faustian offer made to the geometer, since it provides powerful tools, but at the risk of eroding geometric intuition. Crossed modules and hypercrossed complexes are examples of such tools. This project represents a strategy to cheat the devil, by developing these tools in a context in which the geometric intuition is retained.

Research in 2002 and early 2003

The research of Scott Russell Johnson Fellow, Dr Michael Batanin, has attained a heightened level of international recognition, particularly amongst the algebraic topologists. Here is a quote from an email to Street from Professor Jim Stasheff (a pioneer of modern homotopy theory, based at the University of North Carolina): "How could [Michael Batanin] have been unappreciated for so long? Blessings on you for providing all you have for him, and thereby for us."

Batanin has greatly developed higher dimensional homological algebra and the role of higher categorical structures in deformation theory. The biggest achievement connects the classical theory of symmetric operads with higher operads. A beautiful formula for the suspension of a higher operad generalises the Eckmann-Hilton argument to all dimensions, an argument that proved so fundamental in dimensions one and two. The formula also led Batanin to a description of iterated n-fold loop spaces as algebras for contractible n-operads. An application was a description of the combinatorics of iterated loop spaces, which throws light on a 40 year old mystery; this is a subject of further discussion with Getzler, Markl and Tamarkin.

The quality of Batanin's achievements can also be measured by the invitations he accepted in 2002. He participated in the Newton Institute Workshop on Modern Homotopy Theory (Cambridge, September 2002), took up a scholarship in the Max Plank Institute (Bonn, September-November 2002), and gave three invited talks in Bonn and Heitingen Universities (October 2002). He is currently travelling in response to invitations by A. Joyal (Montréal, March-May 2003); P. May (Chicago, April 2003); B. Toen, C. Simpson and A. Hirschowitz (Nice, May 2003); and V. Vershinin (Montpellier, May 2003). He was also invited to lecture at Northwestern University on n-operads; this generated a lot of interest amongst the people doing deformation quantisation.

At the Macquarie University Graduation Ceremony on 10 April 2002 Mark Weber's PhD degree was conferred for the thesis entitled Symmetric Operads for Globular Sets.

At the Macquarie University Graduation Ceremony on 16 April 2003 Lee Flax's PhD degree was conferred with a Vice-Chancellor's Commendation for the thesis entitled Algebraic aspects of entailment. The thesis was also a close runner-up for the Australasian Distinguished Doctoral Dissertation Award in Computer Science for 2002.

The thesis of our Scott Russell Johnson Scholar Daniel Steffen is currently made of three parts. He has described, jointly with James Dolan, a free cartesian closed category in terms of games and "take-back" strategies. The second part throws light on cohomology with coefficients in higher groupoids. The final part picks up on the new topic of "quantum categories" as defined by Day and Street.

Day and Street believe their discovery that star-autonomy is the appropriate concept of duality required to quantize the theory of groupoids is quite exciting. Apart from their appearance in mathematics, star-autonomous categories have been known for over a decade to model the linear logic of Girard used extensively in computer science as a way of coping with resources and resource control. So their appearance in quantum groupoid theory confirms the concept as fundamental. Feedback on this work has been very pleasing.

Ross Street, Max Kelly and Michael Johnson are the three Chief Investigators of the project Category theory arising from geometry, algebra, computer science and physics funded by the Australian Research Council for 2001-3. Around 23 of the first 38 publications listed in the Centre of Australian Category Theory Report for 2002, have arisen on this project, often written jointly with each other and with other colleagues. Each paper contains significant results and original fundamental insights in the areas of the project. For example, the paper with Dr Julien Bichon (France) adapts a known technique for structurally solving the Yang-Baxter system of equations to the solving of a variety of other non-linear systems involving a large number of equations and variables. The paper with Stephen Lack vastly improves one of Street's popular early papers and introduces a more general concept of distributive law between monads with applications in quantum group theory and cohomology. The work with Brian Day and Paddy McCrudden gives a common setting for dualization in monoidal categories and antipodes in bialgebras leading to more-applicable results in the theory of representations of bialgebras. Three papers of Kelly have recently completed the publication process including the paper with Street, Labella, and Schmitt which introduces the new concept of a category enriched on both sides. Kelly is actively working on five other papers that are at an advanced stage. A large part of the work of Johnson under this grant was in fact preliminary mathematical studies which led to the discovery of new techniques for database specification and interoperation. Those studies have been very successful and no fewer than seventeen papers of Johnson were in some way supported by this grant.

Technically, Alexei Davydov discovered the structure of a Gerstenhaber algebra on the K-theory of a braided monoidal category allowing him to provide a truly conceptual and clear proof of the Vafa theorem, important in physics. He furthermore constructed an action of a categorical analogue of the little n-cubes operad on the category of extensions in a monoidal abelian category; this is an interesting weak form of the generalised Deligne hypothesis. The Gerstenhaber bracket he constructs on the endo-extension algebra of the unit object is trivial in the presence of a braiding; yet there is another bracket of degree two that comes into play!

Davydov has discovered other applications of classical homological algebra to homotopy theory. For example, he has a criterion for the "formality" of an A_{infinity}-map in terms of the vanishing of certain Hochschild cohomology groups.

Lack pursued the strong analogy between using pseudo structures and working "up to homotopy'". He showed that the category of 2-categories has a Quillen model structure. This means one can develop a large amount of the homotopy theory of spaces. There are fascinating connections between the resulting homotopy theory and various aspects of 2-category theory.