The Centre of Australian Category Theory provides an algebra of widespread applicability for the synthesis and analysis of systems and processes in fields as diverse as physics and computer science, and also mathematics itself. Established formally in 1999, the Centre consolidates developments including the Australian Category Seminar series, which has been held since 1971.

Vision

To provide the environment for the top international centre for Higher-dimensional Category Theory and a major international centre for general Category Theory.

Mission

To pursue vigorously research into those parts of mathematics and computer science which find natural expression and advancement in terms of Category Theory and to train the highest-quality research students.

Advisory Board

Professor Ross Street	Director CoACT
Professor Michael Johnson	Associate Director CoACT
Emeritus Professor Max Kelly	University of Sydney
Dr Wesley Phoa	Capital Strategy Research, USA
Associate Professor Dominic Verity	Division of ICS, Macquarie University
Professor Di Yerbury	Vice Chancellor, Macquarie University
Professor Jim Piper	Head, Division of ICS, Macquarie University

The Advisory Board met on 18 April 2001 and will meet again on 1 May 2002.

Research in 2001

• Scott Russell Johnson Fellow, Dr Michael Batanin, continued his research on higher-dimensional category theory inspired largely by his expertise in homotopy theory. As reported last year, his work is having a large influence on a number of researchers throughout the world. To provide an explicit example we point to the Peter May Seminar at the University of Chicago which is attended by Fields Medallist Vladimir Drinfeld and other top mathematicians. John Baez from UC@Riverside was invited to explain Batanin's work to that Seminar. Here is Baez's amusing account: Meanwhile, homotopy theory is beginning to creep into lots of subjects, so more people are trying to learn about it. Physicists are getting interested in applying techniques borrowed from homotopy theory to string theory and deformation quantization: . . . Algebraic geometers, on the other hand, have been trying to learn homotopy theory ever since Vladimir Voevodksy used ideas from it to crack a famous open problem: . . . Perhaps for this reason, May has been been explaining homotopy theory to the algebraic geometers Beilinson and Drinfeld, in a seminar which Drinfeld runs "in the Russian style" &emdash; meaning that he asks lots of questions and you talk until you drop dead from exhaustion. I went there and tried to sell them on n-categories, and got quizzed for hours. It was a bit scary, but actually lots of fun. Much better than those seminars where everyone runs for the door after an hour regardless of whether they understood a word - and probably faster, the less they understood!
• Batanin considers his new results (see [45]) on the relationship between his own higher operads and the classical symmetric operads of Peter May to be his best work ever. This is a breakthrough in higher category theory, creating applications in algebraic topology, deformation quantization, and the emerging field of higher homological algebra.
• The current research of our Scott Russell Johnson Scholar Daniel Steffen towards his PhD grew out of preliminary ideas of James Dolan (UC@Riverside). It describes a free cartesian closed category in terms of games and "take-back" strategies.
• Street's work on an ARC Grant 2000-2 for the project Higher dimensional functorial algebra continues to produce publications. The project is funded at $64,000 per annum.
• Max Kelly's work on an ARC Grant 1999-2001 for the project Two-dimensional universal algebra: an investigation of categories with structure produced many publications. The project was funded at $30,000 per annum. The project also includes the ARC Research Fellowship (ARF) which supports Dr Stephen Lack at the University of Sydney; this runs until the end of 2002 since it has been supplemented by funds for one quarter teaching.
• Research conducted by Michael Johnson during his six-month sabbatical leave in 2000 resulted in many publications appearing since.
• As part of his trip in July/August 2001, George Ivanov spoke at the Second Honolulu Conference on Abelian Groups and Modules (see [38] which will appear in the Proceedings). He then moved on to Campinas, Brazil to work with a colleague and lecture on "Rings with infinite sets of orthogonal idempotents".
• It is interesting that the number of joint papers amongst members of CoACT is increasing. International collaboration continues at its usual high rate.

Grants, funding and awards

• The Centre has been actively supported by both Macquarie University and the University of Sydney. In 1998, Mr Steve G Johnson made a very generous donation, in memory of Scott Russell Johnson, a former graduate student in category theory. This has provided for Research Fellowships and Graduate Scholarships.
• The Scott Russell Johnson Memorial donation to CoACT is invested along with the general University funds. The low interest rate averaging around 4.33% in 2001 meant that the available funds sufficed for little more than the salary of the SRJ Fellow.
• Michael Batanin and Alexei Davydov were awarded competitive Scholarships to visit the Max Planck Institute in Bonn, Germany, for a couple of months. They will take these up in mid-2002.
• Michael Batanin and Alexei Davydov were awarded $8000 as a Safety Net Grant. Unfortunately in this round the Batanin/Davydov 2002 ARC Discovery proposal, Davydov's proposed 2002 Australian Research Fellowship and 2002 ARC Discovery Project, and Street's Federation Fellowship proposal were unsuccessful.
• Street was one of six proposed Directors for an ambitious project organized by Peter May for major United States National Science Foundation funding of the project Explorations of higher categorical structures and their applications involving 45 mathematicians from 13 countries. Despite glowing referees reports that remarked on the strength and diversity of the participants, the proposal was not yet considered to be appropriate for funding as a Focused Research Group (FRG) in the Mathematical Sciences.
• Dr Daniel Stevenson (from the University of Adelaide) was awarded a Macquarie University Research Fellowship under the supervision of Street, however, he had already accepted an ARC Fellowship to stay in Adelaide and so declined the MURF.
• Dr Stephen Lack continued his ARC postdoctoral fellowship The algebra of categories with structure under the supervision of Kelly at the University of Sydney. He has submitted a very strong application for a 2003 ARC ARF/QEII Fellowship Two-dimensional algebra: a unified study of structures borne by categories to be tenable at Macquarie University.
• Street applied for a 2003 Discovery Grant entitled Invariants of higher-dimensional categories, with applications, and a Federation Fellowship project entitled Geometric Categorical Algebra. Michael Johnson applied for a 2003 Discovery Grant entitled Categorical universal algebra and foundations of Information Management. CoACT is honoured by the fact that Professor George Janelidze applied for a 2003 Australian Professorial Fellowship tenable at Macquarie University.

Personnel matters

• I am very sad to report that Murray Adelman died on Monday 15 April 2002 as the result of a heart attack. Murray joined the Macquarie University Mathematics Department in 1975. He took early retirement in 1998 and moved to Blackheath with his wife Mary Knowler who died on 26 June 2000. As a PhD student of Peter Freyd, Murray found an elegant construction of the free abelian category. He continued research in category theory and was a stimulating colleague. His recent work with John Corbett on applying topos theory to quantum mechanics is receiving a great deal of attention.
• Michael Johnson was promoted to Professor of Mathematics and Computing in the Division of Information and Communication Sciences, associated with his role as a Director of the Information and Communication Technology Innovations Centre.
• Dominic Verity was promoted to Associate Professor and appointed Director of the Postgraduate Professional Development Program in the Division of Information and Communication Sciences.
• Mark Weber successfully completed his PhD. His Graduation Ceremony was on 10 April 2002. Mark will take up a Postdoctoral Fellowship at the University of Ottawa, Canada, in September 2002.
• Unfortunately, Macquarie University missed the opportunity in 2001 to build on its strengths: it did not appoint outstanding researchers Lack and Janelidze to the positions in the Mathematics Department for which they applied.

Editorial work

• Carboni, Janelidze and Street completed their editorial work on the Special Volume of the Journal of Pure and Applied Algebra dedicated to Professor Max Kelly on the occasion of his 70th birthday. The 17 accepted papers, totalling 396 pages in manuscript form (probably 4 issues of the Journal) are of a pleasingly high quality.
• Street remains editor for six international journals. In particular, his workload for Advances in Mathematics has steadily increased.

Conferences

• Street and Davydov were invited speakers for the Workshop on Noncommutative Geometry, K-theory, and String Theory at the University of Adelaide in Jan 2002. Street gave two lectures under the general title of "Basic structures for higher-dimensional algebra"; while the first lecture dealt with "Substitudes and convolution" and the second, "Higher dimensional categories". Davydov's title was "K-theory of braided monoidal categories". All three talks were very well received by the mathematicians of diverse backgrounds.
• Davydov actively participated in the Singapore International Symposium on Topology and Geometry, Singapore (2-6 Jul 2001) on the way to conducting research at the Max-Planck Institut für Mathematik, Bonn, Germany (8 Jul - 30 Aug 2001).
• Daniel Steffen was successful in 2001 (as in 2000) in being Macquarie's graduate student representative for the 2001 Apple conference in San Francisco.
• The Australian Category Seminar, held each Wednesday afternoon, owes its success in 2001 to the wonderful efforts of the organizers Stephen Lack and Daniel Steffen. Talk titles and abstracts can be obtained here.

Visitors in 2001 and early 2002

James Dolan	University of California @ Riverside	Jul 2000 &endash; Apr 2001
Dr Bachuki Mesablishvili	Georgian Academy of Science	Aug 2000 &endash; Jan 2001
Prof Heinrich Kleisli	Université de Fribourg, Switzerland	Oct 2000 &endash; Apr 2001
Prof George Janelidze	Universidade de Aveiro, Portugal and Mathematical Institute of the Academy of Science, Tbilisi	Nov 2001 &endash; Mar 2002
Prof Bob Rosebrugh	Mt Allison University, Canada	May &endash; Jun 2001 and Mar 2002
Prof Yefim Katsov	Hanover College, Indiana	Apr 2002
Dr Jürgen Koslowski	Technische Universität Braunschweig, Germany	Mar 2002
Prof Aurelio Carboni	Università degli Studi dell'Insubria, Como, Italy	Dec 2001 &endash; Jan 2002
Dr Anne Heyworth	University of Leicester	Feb 2002
Dr Kostas Tourlas	University of Edinburgh	Jan 2002
Prof Marta Bunge	McGill University, Montréal	Jun &endash; Dec 2001
Dr Dusko Pavlovic	Kestrel Institute, Palo Alto, USA	Dec 2001

Documented Research Activity since early 2001

PhD Thesis

1. Mark Weber, Symmetric Operads for Globular Sets (Macquarie University, 2001).

Published

2. S. Lack and E.M. Vitale, When do completion processes give rise to extensive categories? J. Pure Appl. Alg. 159 (2001) 203&endash;230.

3. J.R.B. Cockett and S. Lack, The extensive completion of a distributive category, Theory Appl. Categ. 8 (2001) 541&endash;554.

4. G.M. Kelly and S. Lack, V-Cat is locally presentable or locally bounded if V is so, Theory Appl. Categ. 8 (2001) 555&endash;575.

5. G. Janelidze and G.M. Kelly, A note on actions of a monoidal category, Theory and Applications of Categories 9 (2001) 61&endash;91.

6. J.R.B. Cockett and S. Lack, Restriction categories I: categories of partial maps, Theoretical Computer Science 270 (2002) 223&endash;259.

7. G.M. Kelly, A. Labella, V. Schmitt and R. Street, Categories enriched on two sides, J. Pure Appl. Algebra 168 (2002) 53-98.

8. P. McCrudden, Tannaka duality for Maschkean Categories, J. Pure. Appl. Algebra 168 (2002) 265&endash;307.

9. Alexei Davydov and A. S. Dzhumadil'daev, Factor-complex for Leibniz cohomology. Comm. Algebra 29 (9) (Special issue dedicated to Alexei Ivanovich Kostrikin, 2001) 4197&endash;4210.

10. Jon Berrick and Alexei Davydov, Splitting of Gysing sequence. Algebraic and Geometric Topology 1 (paper no. 37; 2001) 743&endash;762.

11. Michael Johnson and Robert Rosebrugh, Database Interoperability Through State Based Logical Data Independence, International Journal of Computer Applications in Technology, 16 (2-3) (2003).

12. Robert Colomb, C.N.G. Dampney and Michael Johnson, The use of category-theoretic fibration as an abstraction mechanism in information systems, Acta Informatica, 38 (1) (2001) 1&endash;44.

13. Michael Johnson and Robert Rosebrugh, Update algorithms for the sketch data model, Proceedings of the Fifth International Conference on Computer Supported Cooperative Work in Design, London, Ontario, (2001) 367&endash;376.

14. C.N.G. Dampney, Michael Johnson and Robert Rosebrugh, View updates in a semantic data modelling paradigm, Proceedings of the Twelfth Australasian Database Conference ADC2001 (IEEE Press, 2001) 29&endash;36.

15. Michael Johnson and Robert Rosebrugh, View Updatability Based on the Models of a Formal Specification. LNCS, 2001, (2001) 534-549.

16. Michael Johnson and Robert Rosebrugh, Coproducts in Categorical Information System Specification, Proceedings of SCI2001 14 (2001) 145&endash;150.

17. Michael Johnson, Robert Rosebrugh and Richard Wood, Entity Relationship Attribute Designs and Sketches. Theoy and Application of Categories Volume 10 (2002) 94&endash;112.

18. C.N.G Dampney and Michael Johnson, Enterprise Information Systems: Specifying the links among project data models using category theory, Proceedings of the International Conference on Enterprise Information Systems (2001) 619&endash;626.

19. C.N.G Dampney and Michael Johnson, Half-duplex interoperations for cooperating information systems. In Advances in Concurrent Engineering, IN-1, (International Institute of Concurrent Engineering, ISBN 09710461-0-7, 2000) 17pp.

20. Michael Johnson and Robert Rosebrugh, Reverse Engineering Legacy Information Systems for Internet Based Interoperation, Proceedings of the IEEE Conference on Software Maintenance, (2001) 32&endash;39.

21. C.N.G Dampney and Michael Johnson, On Category Theory as a (meta) Ontology for Information Systems Research. In Formal Ontology in Information Systems (edited by Chris Welty and Barry Smith, ACM Press, 2001) 59&endash;69.

22. C.N.G Dampney, Graham Pegler and Michael Johnson, Harmonising Health Information Models - a critical analysis of current practice. Proceedings of the Health Informatics Conference 2001 (ISBN: 0 9585370 8 9) 8pp.

23. Michael Johnson and Robert Rosebrugh, Sketch Data Models, Relational Schema and Data Specifications, ENTCS 61 (6) (2002) 1&endash;13.

24. Michael Johnson and Robert Rosebrugh, Semantics: Initial, Absolute, and Relative, Proceedings of the Tenth OOPSLA Workshop on Behavioral Semantics, Tampa, Florida, October 2001, 121-132.

25. W.P. Joyce, Diagram projection rules for recoupling diagrams in the Racah&endash;Wigner category, J. Math. Physics 42 (2001) 1346-1363.

26. E. Blagoveshchenskaya, G. Ivanov, P. Schultz, The Baer-Kaplansky theorem for almost completely decomposable groups, in: "Abelian groups, rings and modules (Perth, 2000)" Contemp. Math. 273 (Amer. Math. Soc., Providence, RI, 2001) 85&endash;93.|

27. G. Ivanov, Character modules and endomorphism rings of modules over Artinian serial rings, in: "Abelian groups, rings and modules (Perth, 2000)" Contemp. Math. 273 (Amer. Math. Soc., Providence, RI, 2001) 177&endash;189.

Accepted

28. M. Batanin, On the Penon method of weakening of algebraic structures, J. Pure Appl. Algebra.

29. B.J. Day and R. Street, Lax monoids, pseudo-operads, and convolution, Contemporary Mathematics.

30. J. Bichon and R. Street, Militaru's D-equation in monoidal categories, Applied Categorical Structures.

31. R. Street, Functorial calculus in monoidal bicategories, Applied Categorical Structures.

32. B.J. Day, P. McCrudden and R. Street, Dualizations and antipodes, Applied Categorical Structures.

33. R. Street, Weak omega-categories, Contemporary Mathematics.

34. J.R.B. Cockett and S. Lack, Restriction categories II: partial map classification, Theoretical Computer Science.

35. S. Lack and Ross Street, The formal theory of monads II, J. Pure Appl. Alg.

36. S. Lack, Codescent objects and coherence, J. Pure Appl. Alg.

37. J. Adamek, F. Borceux, S. Lack and J. Rosicky, A classification of accessible categories, J. Pure Appl. Alg.

38. George Ivanov and Peter Vamos, A characterization of FGC rings, Contemporary Math. (Amer. Math. Soc. 2002)

Submitted

39. S. Lack, A Quillen model structure for 2-categories.

40. Marta Bunge and S. Lack, A van Kampen theorem for toposes.

41. B.J. Day and R. Street, Abstract substitution in enriched categories.

42. A. Davydov and R. Street, Roots of unity as a Lie algebra.

43. P.H. Butler, W.P. Joyce, L.F. McAven and B.G. Searle, Recursive calculation of non-primitive coupling and recoupling brakets (preprint, April 2001).

44. William Joyce, Natural Associativity without the Pentagon condition.

In Preparation

45. Michael Batanin, Internal operads and Eckmann-Hilton argument (expect to submit in May 2002).

46. Michael Batanin, Computads and Eckmann-Hilton argument (working title).

47. Michael Batanin and Mark Weber, Multitensors and higher dimensional operads.

48. Michael Batanin, Coherence, cooperative games and shuffle polytopes.

49. Michael Batanin, Brian Day and Ross Street, Lax globular monoidal functors out of W. (working title).

50. Michael Batanin, Clemens Berger, and Sjoerd Crans, Contractibility of the operad for Crans' 4-categories (working title).

51. Alexei Davydov, K-theory of braided categories.

52. Alexei Davydov, Nuclei for pseudo-monoidal categories.

53. Alexei Davydov, Homological algebra in monoidal categories.

54. Michael Johnson, Rewriting techniques and coherence theorems (Working document, January 2001).

55. Anne Heyworth and Michael Johnson, Logged rewriting and finite derivation type for monoids (Working document, September 2000).

56. Michael Johnson, David Naumann and John Power, Category Theoretic Models of Data Refinement (Manuscript, 18pp, July 2001).

57. Michael Johnson and Robert Rosebrugh, Universal view updatability (Manuscript, December 2001).

58. S. Crans, G.M. Kelly and A.J. Power, Flexibility for 2-monads.

59. G.M. Kelly and M.-C. Pedicchio, On one-sortedness of algebraic categories.

60. G.M. Kelly and A.J. Power, Enrichment for monads on the category of categories.

61. F. Borceux and G.M. Kelly, On accessibility for enriched categories.

62. B.J. Day and G.M. Kelly, On categories with a distributive law.

63. C. Kennett, M. Roy and M. Zaks, When does skeletal imply coskeletal?