Centre of Australian Category Theory (CoACT)

A Macquarie University Research Centre

Annual Report for 2005

I. List of members

Advisory Board

Professor Ross Street Director of CoACT

Professor Michael Johnson Associate Director of CoACT

Emeritus Professor Max Kelly University of Sydney

Professor Ray Offen (ex officio) Dean, Division of ICS, Macquarie University

Dr Wesley Phoa Capital Strategy Research, USA

Associate Professor Dominic Verity Director, Postgraduate Coursework Programs,

Division of ICS, Macquarie University

Professor Steven Schwartz (ex officio) Vice-Chancellor, Macquarie University

Other Members

Dr Michael Batanin Scott Russell Johnson Memorial Fellow, Math Dept

Dr John Corbett Senior Research Fellow, Math Dept

Dr Alexei Davydov Research Fellow, Math Dept

Dr Brian Day Research Associate, Math Dept

Dr Lee Flax Senior Lecturer, Computing Dept

Ms Carolyn Kennett Director, Macquarie University Numeracy Centre

Dr Steve Lack Senior Lecturer, School of Quantitative Methods and Mathematical Sciences, University of Western Sydney

Ms Catherine Menon PhD Student, Computing Dept

Mr Jon Cohen PhD Student, Computing Dept

Mr Elango Panchadcharam PhD Student, Math Dept

Dr Simona Paoli Australian Postdoctoral Fellow, Math Dept (from July 2005)

Dr Thorsten Palm Scott Russell Johnson Memorial Fellow, Math Dept

Mr Craig Pastro PhD Student, Math Dept

Mr Daniel Steffen PhD Student, Math Dept

Dr Frank Valckenborgh Macquarie University Research Fellow, Math Dept

Visitors since January 2005

Professor Richard Wood Dalhousie University (Dec 2004 – Feb 2005)

Professor Aurelio Carboni Università degli Studi dell'Insubria, Como, Italy (Jan 2005)

Dr Bertrand Toen University of Toulouse (April 2005)

Professor Robin Cockett University of Calgary (April – Aug 2005)

Dr Vincent Schmitt University of Leeds (25 March – 6 April 2005)

Professor André Joyal Université du Québec à Montréal (July 2005)

Professor Clemens Berger Université de Nice Sophia-Antipolis (June – July 2005)

Denis-Charles Cisinski L'Institut Galilée-Université Paris 13 (July – August 2005)

100 StreetFest participants; there were 72 researchers from overseas.

Selected 2005 Highlights

1. StreetFest <http://streetfest.maths.mq.edu.au>: probably the longest ever cutting-edge research conference honouring a scientist’s 60^th birthday, running 11–16 July at Macquarie University and 18–21 July at the Australian National University. An outstanding accomplishment by the organizers: Batanin, Davydov, Johnson, Lack (all CoACT) and Amnon Neeman (ANU).

2. Many excellent contributions are flowing in for the StreetFest Proceedings in preparation for publication by the American Mathematical Society.

3. Of the 100 participants in StreetFest, 72 were overseas researchers. All 64 lectures over 10 days revealed the depth and breadth of CoACT’s research domain.

4. A 2006-8 ARC Discovery Grant was awarded to Street as Chief Investigator for the project Categorical structures in string theory which has thereby become the principal research topic of Davydov.

5. Dominic Verity’s paper on Complicial sets was accepted for Memoirs of the American Mathematical Society.

6. Johnson and Street were on the Scientific Committee for the CT04 Meeting at the University of British Columbia, Vancouver. Street is an editor for the Proceedings volume still in production.

7. Lack and Street are on the Scientific Committee for the CT06 Meeting to be held at White Point, Nova Scotia, 26 June – 1 July 2006.

8. Lack was promoted to Senior Lecturer.

9. Margaret Mitchell completed her PhD (supervised by Johnson) and now works in Silicon Valley.

10. Simon Byrne wrote his Mathematics Honours Essay on Category Theory and will receive a University Medal.

11. Lecture notes from a 1990 Macquarie postgraduate course taught by Street were accepted for publication by Cambridge University Press as a book, entitled Quantum Groups: an entrée to modern algebra, in the Australian Mathematical Society Lecture Series.

II. Short account of each researcher's contribution

The principal business of CoACT is research based in the area of mathematics called Category Theory.

1. Ross Street oversees all operations of CoACT: administration, finance and research.

2. Michael Johnson supports all operations, particularly financial. His research emphasizes applications to computer science.

3. Max Kelly, as patriarch of Australian category theory, researches actively in enriched category theory and 2-dimensional universal algebra, and strongly supports the Australian Category Seminar (ACS).

4. Wesley Phoa is a highly valued industry link with a Cambridge University PhD in category theory.

5. Dominic Verity provides advice in accord with his experience in the finance industry and contributes deeply to research in higher category theory.

6. Michael Batanin is a star researcher in all aspects of CoACT’s research and is strongly motivated by his experience in homotopy theory research.

7. John Corbett is primarily a mathematical physicist using category theory (particularly topos theory) as a foundational tool.

8. Alexei Davydov is a brilliant algebraist who develops the theory of monoidal (bi)categories motivated by his knowledge of group representation theory. He is the organizer of the ACS.

9. Brian Day is an experienced mathematician actively researching enriched category theory — a subject he has deeply influenced since its inception — with inspiration from topological algebra.

10. Carolyn Kennett supports the ACS and her categorical research stems from the theory of simplicial sets.

11. Steve Lack is a brilliant category theorist and is a Chief Investigator of one of our ARC Discovery projects. He is a long-time organizer of the ACS and maintains the invaluable Web page <http://www.maths.usyd.edu.au/u/AusCat/>.

12. Thorsten Palm and Simona Paoli are highly original contributors to higher category theory and are strong supporters of the ACS.

13. Frank Valckenborgh uses category theory in physics; he collaborates with John Corbett and supports the ACS.

14. Catherine Menon and Jon Cohen are PhD students of Michael Johnson.

15. Elango Panchadcharam, Craig Pastro and Daniel Steffen are PhD students of Ross Street.

More detailed research contributions are made explicit in the later sections.

III. Summary of research projects undertaken

While all the papers prepared for publication (listed in Section VI) are projects in their own rights, those modules represent progress towards the greater goals covered by the following grant and personal projects.

Title: 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, Thorsten Palm, George Janelidze, John Corbett, Frank Valckenborgh, 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.

Our research on the project in 2005 was well on track. There were 11 publications appearing and 6 more accepted representing the primary permanent measure of our scientific achievement. Added to this the activity of our Australian Category Seminar can be gleaned from its web site <http://www.maths.usyd.edu.au:8000/u/stevel/auscat/>.

A large number of international linkages and connections were established at a deep level in connection with StreetFest. See <http://streetfest.maths.mq.edu.au/>.

Street’s research on the project involved joint works with a variety of people: Brian Day, Eduardo Dubuc, André Joyal, and Street’s two postgraduate students Elango Panchadcharam and Craig Pastro. Two papers involving Day and Panchadcharam on centres of monoidal categories were submitted; also, a conference paper appeared (see under E2 of Section 5 below). Dubuc presented joint work with Joyal and Street in Amiens, France (see Section 5, item 3 of E5 below).

Kelly’s paper [7] with Vincent Schmitt answers a significant aspect of the project concerning classes of colimits. His papers with Borceux and Janelidze concern a phenomenon that arises in cerrtain semi-abelian categories: namely a generalization of the concept of semi-direct product familiar in the category of groups. The phenomenon is examined in [5], and conditions on a semi-abelian category that guaranty its existence are given in [6].

Johnson’s work with Catherine Menon reached publication status. His work with Bob Rosebrugh on practical database theory is attracting pleasing attention and comment; see IX Items 11 and 12.

Lack’s paper [2] describes the notion of adhesive category, which is a variant of the well-known concept of extensive category. This new notion has been enthusiastically adopted by the graph transformation community. Paper [3] develops some basic 2-category theory which has applications to monoidal comonads. Paper [4] develops the theory of operads in certain sorts of essentially algebraic categories; the main applications are to homotopy theory.

Title: Invariants of higher-dimensional categories, with applications

Personnel: Ross Street (CI), Alexei Davydov.

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.

A final report on this project will be submitted soon. The project proceeded as planned for 2005 and achieved further goals. The funds were used primarily to employ Dr Alexei Davydov as a research fellow.

Davydov proved that the cohomology of a group G with coefficients in a braided crossed G-algebra has the structure of a Gerstenhaber algebra. He also related this with the well-known Gerstenhaber bracket on the Hochschild cohomology; see

¨ Davydov, Cohomology of crossed algebras, Contemporary Mathematics 391 (2005) 41-47.

For a group G, he also defined the new notions of Gerstenhaber G-algebra and of Batalin-Vilkovisky G-algebra. He proved that the K-theory of a crossed braided G-category of a certain type is a Gerstenhaber G-algebra. If the category is balanced (in the sense of Joyal-Street) then its K-theory is a Batalin-Vilkovisky G-algebra; see

A. Davydov and V. Turaev, K-theory of braided crossed G-categories (in preparation).

The notion of chorded (also known as infinitesimally braided) category, which is an infinitesimal analog of braided tensor category, will be investigated. Introduced by Drinfeld as a natural environment for the deformation of classical representation theory, chorded categories appear to be extremely useful in several related areas, in particular, in low dimensional topology where they provide insight into Vassiliev knot invariants.

Elango Panchadcharam, as Street’s PhD student, is studying Mackey functors: these provide invariants for monoidal bicategories constructed from finite groups. A paper

¨ E. Panchadcharam and R. Street, Mackey functors on lextensive categories

is nearing completion and will form a major component of the first author’s PhD.

Title: Foundations of higher dimensional homological algebra

Personnel: Michael Batanin (CI)

Summary: Homotopical Mathematics is a term introduced recently to designate a rapidly developing methodology. It is based on the substitution of set theoretical notions by homotopy theoretical notions in a large part of mathematics relevant to geometry and physics. This approach has already produced spectacular applications in algebraic geometry, topology and mathematical physics. Homological algebra lies at the heart of this approach, yet its further development and application require clear and consistent foundations. The intention of this project is to construct such foundations, using methods of Higher Category Theory. As an outcome, proof of important conjectures from both areas will arise naturally.

Title: Higher categorical structures in homotopy theory and homological algebra

Personnel: Simona Paoli (CI)

Summary: This research is aimed at advancing homotopy theory and homological algebra through the use of higher categorical structures. One goal is to interpret cohomology classes in algebraic categories such as the category of commutative algebras. Another goal is to compare the use of established and more recent higher categorical structures –-groups and weak n-categories – as homotopy models. The proposed approach to the interpretation of cohomology classes is simpler than the existing simplicial methods. Weak n-categories are an emerging field, with applications in diverse areas of mathematics. A comparison between the use of weak n-categories and -groups as homotopy models is unknown, and is imperative in view of applications.

The research project during 2005 proceeded as planned. Paoli tackled the comparison between Tamsamani’s weak n-groupoids and -groups for n = 3 and successfully completed it in this case. This led to interesting results; in particular, it emerged that the comparison problem leads naturally to the study of semistrict higher categorical structures in Tamsamani’s model, making the project even more interesting as little is known about semistrictification in general, and the issue has never been tackled before within Tamsamani’s model. The results relative to the comparison problem for n = 3 are being written up in the following paper:

S.Paoli, From -groups to Tamsamani’s weak 3-groupoids, (in preparation).

While investigating this problem, various background results were needed about Tamsamani’s weak 2-nerves and bicategories, and in particular it was necessary to use a functor between these two categories. While in the literature such a functor had been defined on objects, it turned out that it had not been defined on morphisms. Paoli soon realized the issue was non-trivial. This has now been achieved, and she is in the process of writing up this result in the following paper:

S. Paoli, On bicategories and Tamsamani’s weak 2-categories, (in preparation).

The main objective for the 2006 is the investigation of the comparison problem between Tamsamani’s weak n-groupoids and -groups in the case n > 3. Very interesting and non-trivial issues will arise. Semistrictification of Tamsamani’s weak n-categories will be investigated besides the weak n-groupoid case.

Title: Categorical universal algebra and the foundations of information management

Personnel: Michael Johnson (CI), Jason Rennie, Catherine Menon.

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.

Title: Higher-dimensional categories involving dendrotopes

Personnel: Thorsten Palm

Summary: Palm’s research on definitions of higher-dimensional categories involving dendrotopes has led him to study the broader notion of polytope and polytopic set. He has compiled a list of elementary constructions and results that should be more widely known among category theorists working in the area. One theorem concerns the representability of categories of polytopic sets, and more generally categories admitting a slice-object construction', as full categories of set-valued functors. It particularly yields a new and much shorter proof of the fact that dendrotopic sets form a presheaf category. Currently he is trying to apply the theorem to a certain full category of infinity-computads. Defined similarly to that of dendrotopic sets, it is as wide as to comprise not only R. Street's parity complexes, M. Johnson's pasting schemes and R. Steiner's directed complexes, but also structures without global acyclicity, such as general simplicial and cubical sets (without degeneracies) and, importantly, dendrotopic sets. Papers are in progress.

Title: Categorical structures in string theory

Personnel: Ross Street (CI), Alexei Davydov, Dominic Verity.

Summary: General relativity and quantum field theory are used in physics to explain all the forces of nature. The most promising candidate for the unification of these two fundamental theories is string theory. String theory has exposed exciting mathematical challenges both on the geometric and algebraic side, and for linking those two sides. Category theory has excelled in expressing such linkages in other fields and our proposal details how we plan to use our categorical expertise in string theory. Our results will then feed back into physics. This project begins in 2006.

Title: Cohomology enhanced: an application of enriched and higher categories

Personnel: Ross Street (CI), Michael Johnson (CI), Stephen Lack (CI), Dominic Verity (CI), Max Kelly.

Summary: ARC Proposal for 2007: Cohomology has been one of the most powerful tools in the mathematics of the twentieth century, finding applications in all areas of modern mathematics. It is a technique for understanding and classifying complex mathematical structures in simpler terms. The current project involves a radical expansion in scope of the information extracted from these mathematical structures, using the most recent advances in enriched and higher-dimensional category theory. Motivated by the needs of physicists, computer scientists, and colleagues in other similar fields, mathematicians study highly complicated structures which are typically hard to understand completely in concrete terms. Cohomology is an invaluable technical tool which allows data to be extracted from these complex structures. This project will involve a radical expansion in scope of the amount and type of data so extracted. This is made possible by the most recent advances in higher-dimensional category theory.

IV. Existing and potential internal and external linkages and collaborative arrangements

Section I makes many linkages apparent by noting the members, their departments, their institutions, and their visitors. CoACT has significant linkages with researchers in Montréal, Halifax, Milano, Chicago, Riverside, Cambridge (U.K.), Louvain-la-neuve, and San Jose (including some companies).

A linkage between CoACT and Professor Amnon Neeman’s group at the Australian National University strengthened during 2005 with the StreetFest Workshop being held there.

V. Progress in relation to agreed performance indicators (A–G below) specified in the Centre's strategic plan

See Appendix 2.

VI. List of Centre publications, materials submitted for publication, provisional patents and other forms of commercialisation, and other measures of research output, including evidence of impact (e.g. citations, uptake of research developments by other groups, media reports, etc).

The Selected 2005 Highlights listed in Section I provide clear evidence of CoACT’s international impact. In particular, we point to the incredible success of StreetFest in July 2005.

Publications appearing in 2005

[1] R. Street, Enriched categories and cohomology with author commentary, Reprints in Theory and Applications of Categories No. 14 (2005) 1–18.

[2] Stephen Lack and Pawel Sobocinski, Adhesive and quasiadhesive categories, Theor. Inform. Appl. 39 (2005) 511–545.

[3] Stephen Lack, Limits for lax morphisms. Appl. Categ. Structures 13 (2005) 189–203.

[4] Stephen Lack and Simona Paoli, An operadic approach to internal structures, Appl. Categ. Structures 13 (2005) 205–222.

[5] F. Borceux, G. Janelidze and G.M. Kelly, Internal object actions, Comment. Math. Univ. Carolin. 46 (2005) 235–255.

[6] F. Borceux, G. Janelidze and G.M. Kelly, On the representability of actions in a semi-abelian category, Theory and Applications of Categories 14 (2005) 244–286.

[7] G M Kelly and V. Schmitt, Notes on enriched categories with colimits of some class, Theory and Applications of Categories 14 (2005) 399–423.

[8] G.M. Kelly, Basic concepts of enriched category theory, Reprints in Theory and Applications of Categories No. 10 (2005), vi+137 pp.

[9] G.M. Kelly, On the operads of J.P. May, Reprints in Theory and Applications of Categories No. 13 (2005) 1–13.

[10] Simona Paoli, On the non-balanced property of the category of crossed modules in groups, J. Pure Appl. Algebra 197 (2005) 19–22.

[11] A. Davydov, Cohomology of crossed algebras, Contemporary Mathematics 391 (2005) 41-47.

Accepted for publication

[12] Dominic Verity, Complicial sets, Memoirs of the American Math. Society (accepted 5 December 2005); preprint at http://arxiv.org/pdf/math.CT/0410412.

[13] M. Batanin, The Eckmann-Hilton argument and higher operads, Advances in Math. (to appear).

[14] Ross Street, An Australian conspectus of higher categories, Proceedings of the 2004 Summer Program: n-Categories: Foundations and Applications, 1-18 June 2004 at the IMA of the University of Minnesota, Minneapolis (accepted July 2005).

[15] Brian Day, Elango Panchadcharam and Ross Street, Lax braidings and the lax centre, Contemporary Mathematics (accepted 3 February 2006).

[16] Brian Day and Ross Street, Centres of monoidal categories of functors, Contemporary Mathematics (accepted 21 February 2006).

[17] Ross Street, Quantum Groups: an entrée to modern algebra, Australian Mathematical Society Lecture Series (Cambridge University Press; accepted 25 February 2006).

Submitted

[18] Thorsten Palm, Categories with slicing (22 page preprint).

[19] J.R.B. Cockett and Stephen Lack, Restriction categories III: colimits, partial limits, and extensivity.

[20] M. Batanin, Computads and slices of operads, http://arxiv.org/pdf/math.CT/0209035; submitted to Theory and Appl. of Categories.

[21] M. Batanin, The combinatorics of iterated loop spaces, http://arxiv.org/pdf/math.CT/0301221 ; submitted to Topology and its Appl.

[22] M. Batanin, Symmetrisation of n-operads and compactification of real configuration spaces; submitted to Advances in Math.

In Preparation

[23] S. Paoli, From -groups to Tamsamani’s weak 3-groupoids.

[24] S. Paoli, On bicategories and Tamsamani’s weak 2-categories.

[25] E. Panchadcharam and R. Street, Mackey functors on lextensive categories.

[26] Eduardo Dubuc, André Joyal and Ross Street, A construction of 2-filtered bicolimits of categories (24 page preprint).

[27] R. Buchweitz, A. Davydov and R. Street, The Gerstenhaber homotopy in a monoidal bicategory.

[28] Ross Street, Monoidal actions, enriched categories, and convolution.

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

[30] A. Carboni, G. Janelidze, G.M. Kelly and Stephen Lack, Pointing a category .

[31] G. Janelidze, G.M. Kelly and Stephen Lack, Factorization systems and Galois theory in the 2-dimensional context.

[32] B.J. Day and Stephen Lack, Limits of small functors.

[33] Michael Batanin and Mark Weber, Multitensors and higher dimensional operads.

[34] Michael Batanin, Coherence, cooperative games and shuffle polytopes.

[35] Michael Batanin, Brian Day and Ross Street, Lax globular monoidal functors out of W.

[36] Michael Batanin, Clemens Berger, and Sjoerd Crans, Contractibility of the operad for Crans' 4-categories.

[37] Alexei Davydov, Nuclei for pseudo-monoidal categories.

[38] Alexei Davydov, Gerstenhaber structures on extensions.

[39] Alexei Davydov, -structures and Hochschild cohomology.

[40] Michael Johnson, Rewriting techniques and coherence theorems.

[41] Michael Johnson, David Naumann and John Power, Category Theoretic Models of Data Refinement.

[42] Michael Johnson and Robert Rosebrugh, Universal view updatability.

[43] Michael Johnson, Half-duplex interoperation.

[44] Michael Johnson, Nulls and opcartesian properties.

[45] Michael Johnson, Constant complements and reversibility.

[46] G.M. Kelly and M.-C. Pedicchio, On one-sortedness of algebraic categories.

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

[48] F. Borceux and G.M. Kelly, On accessibility for enriched categories.

[49] B.J. Day and G.M. Kelly, On categories with a distributive law.

[50] G.M. Kelly and B. Mesablishvili, On enriched monads of descent type and of effective descent type.

[51] Alexei Davydov, Quasi-commutative monoids.

[52] Alexei Davydov, Generators and relations for categories of representations of symmetric groups.

[53] Stephen Lack and Pawel Sobocinski, Quasiadhesive categories, quasitoposes, and Artin gluing.

[54] Stephen Lack and Simona Paoli, On 2-nerves of bicategories.

[55] Dominic Verity, Weak Complicial Sets, Part I: Basic Homotopy Theory.

[56] Dominic Verity, Weak Complicial Sets, Part II: Nerves of Complicial Gray-Categories.

[57] Dominic Verity, Weak Complicial Sets, Part III: Equivalence Stratification.

[58] Dominic Verity, Weak Complicial Sets, Part IV: Internal Quasi-Category Theory.

Conference presentations

[59] G.M. Kelly, Ross Street and the early days of Category Theory in Australia, Presentation at StreetFest (Macquarie University and ANU, 11-21 July 2005).

[60] Ross Street, Centres, Presentation at StreetFest (Macquarie University and ANU, 11-21 July 2005).

[61] 62 further talks at StreetFest; for titles and abstracts see <http://streetfest.maths.mq.edu.au>

[62] B. Day, E. Panchadcharam, and R. Street, On centres and lax centres of promonoidal categories, Paper for the International Conference “Charles Ehresmann: 100 ans” (Amiens, 7-9 October 2005) http://perso.wanadoo.fr/vbm-ehr/ChEh/articles/Street.pdf

[63] E. Dubuc, A. Joyal and R. Street, Bifiltered colimits of categories, Presentation at the International Conference “Charles Ehresmann: 100 ans” (Amiens, 7-9 October 2005).

VII. A financial summary for the year, including details of external grants and contracts, and projected cash flow and budget for the following year.

See Appendix 3.

VIII. The status of implementation of the recommendations in the latest review of the Centre.

Reviewers have been nominated. Soon after submission of this 2005 Report, a report for the last five years and the plans for the future will be produced along with other material in accordance with the Terms of Reference for University Research Centres as supplied by the Research Office.

IX. Proposed activities for the coming year, and related performance indicators.

Plans for 2006 are very well developed. It will be another big year for CoACT. We expect to meet all our benchmarks. Our activities will include the following points.

1. We look forward to a visit by Mr Steve Johnson in the second half of March, the first since his donation and the formal establishment of CoACT.

2. SRJ Fellow Michael Batanin will take up invitations to visit research centres in Strasbourg and Nice (France), Minneapolis and Chicago (USA), and Montréal (Canada) during March–June.

3. Alexei Davydov will visit the Max Planck Institute in Bonn (Germany) and consult with V. Turaev on the String Theory project.

4. We will celebrate the appearance of three significant books: Verity’s Complicial Sets, Street’s Quantum Groups: an entrée to modern algebra, and the StreetFest Proceedings.

5. We are submitting a 2007 ARC Discovery Proposal Cohomology enhanced: an application of enriched and higher categories.

6. The Annual Conference on Category Theory CT06 will be held at White Point, Nova Scotia (Canada) — Lack and Street are on the Scientific Committee.

7. Lack has sabbatical leave for the first half of 2006. We are delighted that he is currently visiting Macquarie University. He will spend the second half of the leave visiting the University of Chicago.

8. Lack is an invited plenary speaker for CT06; Palm, Panchadcharam and Paoli have submitted abstracts for talks; while Kelly, Michael Johnson, and Pastro will also participate.

9. Panchadcharam is applying to the Macquarie University Postgraduate Research Fund (Round 2, 2006) for support to attend CT06 and to take up an invitation to Dalhousie University, while both he and Pastro have applied to the conference organizers for partial financial support available competitively to graduate students; Pastro has applied for Canadian funding for other conferences in the weeks before CT06.

10. In September, Macquarie University Mathematics Department will host the Jubilee Annual Meeting of the Australian Mathematical Society for which Street is on the Plenary Sessions Committee and is Treasurer, while Lack and Paoli will organize the Special Sessions on Category Theory.

11. Batanin is a plenary speaker for Jubilee Australian Math. Society Meeting.

12. Michael Johnson will lead research into category theoretic based systems interoperation by completing papers [43], [44] and [45], by attending I-ESA (Interoperability for Enterprise Software and Applications) in Bordeaux (March 2006), by presenting an address to the International Workshop on Enterprise Interoperability invited by the European Commission, by presentations to the Database theory experts in Edinburgh, and attending and editing the Proceedings for AMAST 06 in Estonia (July 2006).

13. Professor Robert Rosebrugh from Canada will visit in March-April, mainly to work with Michael Johnson who is also working with the Curie Institute (Paris) and Philip Wadler (Edinburgh) on related projects.

14. Professor Robert Coquereaux from Marseille (France) will visit in August and September to consult on applications of monoidal categories to physics.

15. Street has OSP for the second half of 2006 and, along with advancing projects already mentioned, intends to commence writing a book on higher categories.

16. Menon is expected to submit her PhD in mid 2006 while Panchadcharam is expected to submit his PhD thesis later in the year.

17. At least two new PhD students are expected to commence during the year.

18. In July 2006, Mark Weber will join us as a Research Fellow funded by Batanin’s Discovery Grant.

19. The Australian Category Seminar will continue meeting on Wednesday afternoons — speakers, talk titles and some abstracts will continue to be documented on the ACS website: <http://www.maths.usyd.edu.au/u/AusCat/>.

20. Editorial work will continue on track — the StreetFest and CT04 Proceedings should appear.

21. The Review of CoACT will take place, clearing the way for our application for Mature Research Centre status.

Many of these items relate to the agreed performance indicators (A–G as in Appendix 2) as follows:

A. Item15.

B. Items 4 and 12.

C. Item 16.

D. Items 7, 10 and 11.

E. Item 15. International students are consistently approaching our members as PhD supervisors. Street is currently corresponding with 2 strong potential students. Simon Byrne, who wrote his Math. Honours project with Street and received the University Medal, is considering postgraduate work but may choose to do it overseas.

F. Street, Johnstone, Lack and Verity have applied for an ARC Discovery Grant for 2007. Palm has applied for an ARC Postdoctoral Fellowship for 2007 under Street’s supervision. Boris Chorny has applied for an ARC Postdoctoral Fellowship for 2007 under Batanin’s supervision.

G. Reviewing is expected to continue at the same rate. In particular, Street will review the reprinting of Kelly’s book “Basic Concepts of Enriched Category Theory” for Zentralblatt.

Appendix 1: Poster for the Macquarie University part of StreetFest

(designed by Daniel Steffen)

<http://streetfest.maths.mq.edu.au/poster>

Appendix 2: Detail on Section V

Progress in relation to agreed performance indicators (A–G below) specified in the Centre's strategic plan

A. Activity of the Australian Category Seminar (ACS)

Benchmark At least 40 three-hour seminars per annum.

33 three-hour Seminars were held in 2005. For details of dates, speakers, titles and some abstracts, see <http://www.maths.usyd.edu.au/u/stevel/auscat/>. The StreetFest, with its 64 talks, can be viewed as a vast extension of the ACS.

B. Scholarly publication in international refereed journals

Benchmark Fifteen journal articles per annum.

11 publications have appeared in refereed journals in 2005. Another 6 are accepted and another 7 submitted. See Section VI for details.

C. Editing of papers submitted to scholarly journals and to special issue volumes

Benchmark One special issue volume per annum.

Batanin, Davydov, Johnson, Lack and Neeman are editors of the Proceedings of StreetFest in preparation for publication by the American Mathematical Society.

Street is an editor (with John MacDonald, George Janelidze and Walter Tholen) of the Proceedings of CT04 to appear in Theory and Applications of Categories.

Johnson is a founding editor of the International Journal of Mathematics and Computer Science, and Steering Committee Chair of the AMAST initiative.

Batanin and Lack are on the Editorial Board of Applied Categorical Structures and Lack is also on the Board of Theory and Applications of Categories.

Street continued actively on the Editorial Boards of six international journals:

Advances in Mathematics; Applied Categorical Structures; Theory and Applications of Categories; Journal of Homotopy and Related Structures (JHRS previously HHA); Bulletin of the Australian Mathematical Society; Cahiers de topologie et géométrie différentielle catégoriques

Kelly continued actively on the Editorial Boards of: Applied Categorical Structures and Theory and Applications of Categories.

D. International plenary addresses and membership of conference scientific steering committees

Benchmark One international plenary address or one conference steering committee per annum.

1) Batanin, Davydov, Johnson, Lack and Neeman organized the International Conference on Categories in Algebra, Geometry, and Mathematical Physics at Macquarie University (11–16 July 2005) and the follow-up Workshop at the Australian National University (18–21 July 2005).

2) Street is on the Scientific Committee for the annual category meeting CT06 at White Point (Nova Scotia, Canada; 26 June – 1 July 2006).

3) Steve Lack will be an invited speaker at the Conference CT06.

4) Street has agreed to be on the Scientific Committee for the Conference: The Mathematics of String Theory at the ANU in 2007.

E. Effective Full-Time Student Units (EFTSU) attracted and completion rates, especially for postgraduate students

Benchmark Ten postgraduate EFTSU and two PhD completions per annum.

Margaret Mitchell completed her PhD and now works in Silicon Valley. Jon Cohen, Catherine Menon (about to submit her thesis), Elango Panchadcharam, and Craig Pastro are current PhD students. Daniel Steffen is expected to submit his thesis in 2006. Street is awaiting the outcomes of applications by two students for PhD candidature under his supervision.

F. Funding attracted through grant applications and other sources

Benchmark $300000 in external funding per annum.

See VII below: 179% of the benchmark. This high figure is influenced by the special circumstances of the StreetFest.

G. Visibility in the scientific review literature

Benchmark 20 reviews written by CoACT members appearing in Mathematical Reviews, Zentralblatt für Mathematik, or other influential mathematical and computing reviewing journals.

Benchmark exceeded. Around 24 papers were reviewed by Street and Lack for Math. Reviews and Zentralblatt. In particular, for the latter, Street reviewed the first two volumes of the trilogy: “Sketches of an elephant: A topos theory compendium” by Peter Johnstone.

Appendix 3: Detail on Section VII

VI. A financial summary for the year, including details of external grants and contracts, and projected cash flow and budget for the following year.

Financial Summary 2005:

EXTERNAL Grants and Contracts Income

SRJ Fellowship interest 96369.24 (6.05% return)

Street-Kelly-Johnson-Lack ARC 73142.00

Street ARC 74872.00

Morgan Phoa donation 10404.47

StreetFest: AMSI 9181.82

Macq Math Dept 5800.00

ANU for Sydney accom. 26395.00

(approx) ANU for Airfares 20200.00

(approx) ANU for Canberra accom. 10000.00

(approx) Neeman grant contrib. 23800.00

Batanin ARC 51050

Paoli APD 68911

Valckenborgh MURF (est) 68000

TOTAL 538126 (179% of benchmark)

Proposed Budget for 2006:

INCOME

SRJ Fellowship interest (est) 95000

Street-Kelly-Johnson-Lack ARC 70000

Street ARC 95000

Batanin ARC 50000

Paoli APD 69000

Valckenborgh MURF (est) 69000

TOTAL 448000 (149% of benchmark)

PLUS funds carried forward

Operating acct 2047.76

Morgan Phoa donation 26713.59

S-K-J-L ARC 50097.13

Street ARC 38531.18

TOTAL 117389.66

FUNDS AVAILABLE 565390

LESS projected estimated expenditure

SRJ Fellow (Batanin) xxxxx

SRJ Fellow (Palm) xxxxx

MURF (Valckenborgh) xxxxx

ARC Research Fellow (Paoli) xxxxx

Res Fellow (Davydov) xxxxx

Res Fellow (Day) xxxxx

Visitors xxxxx

Travel xxxxx

Outstanding commitments xxxxx

Equipment xxxxx

TOTAL 478000

PROJECTED C/F to 2005 87390

Because of our strong financial position there is no need for more detailed cash-flow analysis for 2006.