**Centre of Australian Category Theory**

**Report for 2002**

The Centre of Australian Category Theory CoACT develops 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. Although having operated as a coherent group since the founding of the on-going Australian Category Seminar in 1971, CoACT was formally established in 1999.

**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 Ray Offen |
Dean, Division of ICS, Macquarie University |

**Honours**

- Both Max Kelly and Ross Street were awarded Australian
Federation Centenary Medals Medals with the citation:
*for service to Australian society and science in mathematics.*Appropos to this, an interview with Kelly appeared in the Sydney Morning Herald (page 4 on 22 April 2003). - A publication dedicated to Professor Max Kelly on the occasion
of his 70th birthday appeared as Volume 175 Numbers 1-3 of the
*Journal of Pure and Applied Algebra.*

**Research in 2002 and early 2003**

- This Report indicates the superb quality of research by members of CoACT for the period.
- Since the last report, the research of Scott Russell Johnson Fellow, Dr Michael Batanin, 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; see [31]. 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 [33] 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 [34].
- Day and Street believe their discovery [34] 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 below 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 [26] 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 [3] 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 [27] 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 [2] 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 [9] 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 [44] 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 [45] 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 [13] 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.
- The visitors to CoACT continue to enrich our research through joint collaboration and through the Seminar; see the list below. Of particular note was the extremely successful visit by Professor John Baez who, whilst here, also gave some beautiful lectures to wider mathematical audiences.

**Grants and funding**

- 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 both Research Fellowships and Graduate Scholarships.
- The Scott Russell Johnson Memorial donation to CoACT is invested along with the general University funds. The interest rate has been slowly increasing. For 2002 it averaged 4.05% p.a. while for January 2003 it was 5% p.a.
- Street's Australian Research Council Discovery Grant
application for 2003-5 was successful. For the project entitled
*Invariants of higher-dimensional categories, with applications,*the allocation was $90K, $60K and $70K over the three years. The funds are being used primarily to support Dr Alexei Davydov as Research Associate on the project. - The Street-Kelly-Johnson ARC Large Grant is in its final year (2003). A continuation of this funding is sought in the form of a 2004-6 Discovery Grant proposal where, we are happy to report, Stephen Lack has joined as a fourth Chief Investigator.
- Michael Johnson's work as a Chief Investigator with Street and
Kelly continues to be very productive. In particular, it led to
the discovery of new techniques for database specification and
interoperation. Carrying this particular direction further would
be beyond the scope of that grant or its proposed continuation.
Consequently Johnson has prepared a separate Discovery Grant
application, entitled
*Category theoretic foundations of information management,*seeking funding for the continuation of that now more applied work. - For the fourth consecutive year Daniel Steffen was awarded a WWDC 2003 Apple University Consortium Competitive Scholarship to be Macquarie's representative at the Apple Computer Conference in San Francisco.
- Michael Batanin and Alexei Davydov took up their competitively awarded Scholarships to visit the Max Planck Institute in Bonn, Germany, each for a couple of months in mid-2002.
- Michael Batanin and Alexei Davydov held a Macquarie University Safety Net Grant during 2002.

**Personnel matters**

- Dr Stephen Lack accepted a position as Lecturer in the School of Quantitative Methods and Mathematical Sciences at the University of Western Sydney.
- Dr Mark Weber will continue for a second year as Postdoctoral Fellow at the University of Ottawa, Canada.

**Conferences**

- As reported last year, 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. All three talks were very well received by the mathematicians of diverse backgrounds. - Street was a plenary speaker at the Fields Institute (Toronto,
Canada) during the
*Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories,*23-28 September 2002; audio and text of his lecture*Formal representation theory*are at <http://www.fields.utoronto.ca/audio/02-03/galois_and_hopf/street/>. The two papers [34] and [35] were written in connection with this Workshop. - Lack gave two talks at the September Fields Institute
*Workshop*: his own, entitled*A van Kampen theorem for toposes*(see [28]) and Kelly's*On flexibility for 2-monads*(see [49]). Although Kelly was an invited speaker, at the last moment he was unable to attend. - Street was invited to speak at Université du
Québec à Montréal in a very lively
international colloquium in honour of his close colleague
Professor André Joyal. The
*Journées Joyal*were 11-13 April 2003; see <http://omega.CRM.UMontreal.CA/Joyal/>. Street's talk, entitled*Quantum categories and quantum groupoids,*was based on the joint work [34] with Brian Day. In the days around*Journées Joyal,*Street worked with Joyal, Dubuc and others in Montréal for the colloquium. At least two germs of possible future research arose. - Batanin is an invited speaker at the Workshop
*Cats2*on higher categorical structures in algebraic geometry in Nice, France (17-24 May 2003). The organizers are A. Hirschowitz, C. Simpson and B. Toen. - There was approval for a Summer Program (7-18 June 2004),
called
*n-Categories: Foundations and Applications,*at the Institute for Mathematics and its Applications (IMA) at Minnesota. The organizers of the program are J. Peter May and John Baez. The webpage (recommended reading!) is <http://www.ima.umn.edu/categories/>. - The
*Australian Category Seminar,*held each Wednesday afternoon, owes its success in 2002 again to the wonderful efforts of the organizers Stephen Lack and Daniel Steffen. Talk titles and abstracts can be obtained from the website: <http://www.maths.usyd.edu.au:8000/u/stevel/auscat/titles.html>.

**Visitors in 2002 and early 2003**

John Baez |
University of California, Riverside |
January-March 2003 |

Isar Stubbe |
Université Catholique de Louvain-la-Neuve, Belgium |
January 2003 |

Eric Goubault |
Commissariat à l'Energie Atomique, Saclay, France |
December 2002 |

Shahn Majid |
Queen Mary, University of London |
December 2002 |

Yefim Katsov |
Hanover College, Indiana |
April 2002 |

George Janelidze |
Universidade de Aveiro, Portugal and Mathematical Institute of the Academy of Science, Tbilisi |
Nov 2001 - Mar 2002 |

Bob Rosebrugh |
Mt Allison University, Canada |
Mar 2002 |

Jürgen Koslowski |
Technische Universität Braunschweig, Germany |
Mar 2002 |

Anne Heyworth |
University of Leicester |
Feb 2002 |

Kostas Tourlas |
University of Edinburgh |
Jan 2002 |

**Documented Research Activity since early
2002**

**Published**

1. M. Batanin, On the Penon method of weakening of algebraic
structures, *J. Pure Appl. Algebra* **172** (2002) 1-23.

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

3. S. Lack and R. Street, The formal theory of monads II, *J.
Pure Appl. Algebra* **175** (2002) 243-265.

4. B.J. Day and R. Street, Lax monoids, pseudo-operads, and
convolution, in: "Diagrammatic Morphisms and Applications",
*Contemporary Mathematics* **318** (American Math Soc. 2003)
75-96.

5. R. Street, Trees, permutations and the tangent function,
*Reflections* **27 (2)** (Math. Assoc. of NSW, May 2002)
19-23; also *arXiv*: math.HO/0303267.

6. R. Street, Weak omega-categories, in: "Diagrammatic Morphisms
and Applications", *Contemporary Mathematics* **318**
(American Math Soc. 2003) 207-213.

7. B.J. Day and R. Street, Abstract substitution in enriched
categories, *J. Pure Appl. Algebra* **179** (2003) 49-63.

8. A. Davydov and R. Street, Roots of unity as a Lie algebra,
*Georgian Mathematical Journal* **9(4)** (2002) 683-690.

9. A. Davydov, K-theory of braided monoidal categories,
*K-Theory* **27** (2002) 371-389.

10. J.R.B. Cockett and S. Lack, Restriction categories I:
categories of partial maps, *Theoretical Computer Science*
**270** (2002) 223-259.

11. J. Adamek, F. Borceux, S. Lack and J. Rosicky, A
classification of accessible categories, *J. Pure Appl. Algebra*
**175** (2002) 7-30.

12. S. Lack, Codescent objects and coherence, *J. Pure Appl.
Algebra* **175** (2002) 223-241.

13. S. Lack, A Quillen model structure for 2-categories,
*K-Theory* **26** (2002) 171-205.

14. J.R.B. Cockett and S. Lack, Restriction categories II: partial
map classification, *Theoretical Computer Science* **294**
(2003) 61-102.

15. A. Carboni, G. Janelidze and R. Street, Foreword [on the 70th
birthday of Gregory Maxwell Kelly]. Special volume celebrating the
70th birthday of Professor Max Kelly. *J. Pure Appl. Algebra*
**175** (2002) 1-5.

16. G. Janelidze, G.M. Kelly, A note on actions of a monoidal
category. *Theory Appl. Categ.* **9** (2001/2), 61-91.

17. P. McCrudden, Tannaka duality for Maschkean Categories, *J.
Pure. Appl. Algebra* **168** (2002) 265-307.

18. Paddy McCrudden, Opmonoidal monads, *Theory and Applications
of Categories* **10** (2002) 469-485.

19. Michael Johnson and Robert Rosebrugh, Database
Interoperability Through State Based Logical Data Independence,
*International Journal of Computer Applications in Technology*
**16 (2-3)** (2003) 97-102.

20. Michael Johnson, Robert Rosebrugh and Richard Wood, Entity
Relationship Attribute Designs and Sketches, *Theoy and Application
of Categories* **10** (2002) 94-112.

21. Michael Johnson and Robert Rosebrugh, Sketch Data Models,
Relational Schema and Data Specifications, *ENTCS* **61 (6)**
(2002) 1-13.

22. C.N.G Dampney and Michael Johnson, Enterprise Information
Systems: Specifying the links among project data models using
category theory, *Enterprise Information Systems III* (Eds J.
Filipe, B. Sharp, and P. Miranda; Kluwer 2002) 191-197.

23. Richard Buckland, Michael Johnson and Dominic Verity, On the
specification of Higher Dimensional Automata, *ENTCS* **68**
(2002) 1-13.

24. Michael Johnson and Rosebert Rosebrugh, Three approaches to
partiality in the sketch data model, *ENTCS* **78** (2003)
1-18.

**Accepted**

25. J. Bichon and R. Street, Militaru's D-equation in monoidal
categories, *Applied Categorical Structures* (to appear).

26. R. Street, Functorial calculus in monoidal bicategories,
*Applied Categorical Structures* (to appear).

27. B.J. Day, P. McCrudden and R. Street, Dualizations and
antipodes, *Applied Categorical Structures* (to appear).

28. M. Bunge and S. Lack, Van Kampen theorems for toposes,
*Advances in Math.* (to appear).

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

30. P.H. Butler, W.P. Joyce, L.F. McAven and B.G. Searle,
Recursive calculation of non-primitive coupling and recoupling
brakets, *Canadian Journal of Physics*

**Submitted**

31. M. Batanin, The Eckmann-Hilton argument, higher operads and
-spaces, math.CT/0207281; submitted to *Advances in Math.*

32. M. Batanin, Computads and slices of operads, math.CT/0209035;
submitted to *Theory and Appl. of Categories.*

33. M. Batanin, The combinatorics of iterated loop spaces,
math.CT/0301221; submitted to *Topology and its Appl.*

34. B.J. Day and R. Street, Quantum categories, star autonomy, and quantum groupoids, math.CT/0301209; submitted to an American Math Soc. publication "Categories, Algebras, and Galois Theory - Proceedings of the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, Toronto 2002";.

35. R. Street, Categorical and combinatorial aspects of descent
theory, math.CT/0303175; submitted to *Applied Categorical
Structures.*

36. R. Street, The monoidal centre as a limit, math.CT/0304053.

37. G.M. Kelly and S. Lack, Monoidal functors generated by adjunctions, with applications to transport of structure.

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

**In Preparation**

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

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

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

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

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

44. Alexei Davydov, Gerstenhaber structures on extensions.

45. Alexei Davydov, A_{infinity}-structures and Hochschild cohomology.

46. Michael Johnson, Rewriting techniques and coherence theorems.

47. Michael Johnson, David Naumann and John Power, Category Theoretic Models of Data Refinement.

48. Michael Johnson and Robert Rosebrugh, Universal view updatability.

49. G.M. Kelly, S. Lack and A.J. Power, Flexibility for 2-monads.

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

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

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

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

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

55. G.M. Kelly and B. Mesablishvili, On enriched monads of descent type and of effective descent type (28 page working note, 2002).

56. C. Dampney and M. Johnson, Experience in developing inteoperations among legacy information systems using partial reverse engineering (Manuscript, February, 2003).