1. The category of opetopes and the category of opetopic sets.  In Theory and Applications of Categories, 11:353–374, 2003.  E-print math.CT/0304284
  2. Pseudo-distributive laws.  With Martin Hyland and John Power, in Electronic Notes in Theoretical Computer Science, 83, 2004.
  3. Weak n-categories: opetopic and multitopic foundations.  In Journal of Pure and Applied Algebra, 186(2):109–137, 2004.  E-print math.CT/0304277
  4. Weak n-categories: comparing opetopic foundations.  In Journal of Pure and Applied Algebra, 186(3):219–231, 2004.  E-print math.CT/0304279
  5. A relationship between trees and Kelly-Mac Lane graphs.  In Mathematical Proceedings of the Cambridge Philosophical Society, 141(1):33-56, 2006.  E-print math.CT/0304287
  6. The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories.  With Nick Gurski. In Categories in Algebra, Geometry and Mathematical Physics, proceedings of Streetfest, editors Batanin et al, Contemporary Math. AMS 431:143-164, 2007.  Also available on the arXiv 0708.1178. Long draft with more explanation and diagrams is available here.
  7. An omega-category with all duals is an omega groupoid.  In Applied Categorical Structures, 15(4):439–453, 2007. Also available here.
  8. Towards an n-category of cobordisms. With Nick Gurski, 2006.  In Theory and Applications of Categories, 18:274-302, 2007. E-print 0708.1178.
  9. Monad interleaving: a construction of the operad for Leinster’s weak omega-categories. In Journal of Pure and Applied Algebra, 214:809–821, 2009. Also available here.
  10. A note on Penon’s definition of weak n-category. (With Michael Makkai). In Cahiers de Topologie et Géométrie Différentielle Catégoriques, 50:83–101, 2009. E-print 0907.3961.
  11. Iterated distributive laws. In Mathematical Proceedings of the Cambridge Philosophical Society, 150(3):459–487, 2011. E-print 0710.1120.
  12. Comparing operadic theories of n-category, 2008, 47 pages. In Homotopy, Homology and Applications, 13(2):217–249, 2011. E-print 0809.2070.
  13. The periodic table of n-categories for low dimensions II: degenerate tricategories.  (With Nick Gurski). 33 pages. In Cahiers de Topologie et Géométrie Différentielle Catégoriques 52:82–125, 2011. E-print 0706.2307.
  14. A direct proof that the category of 3-computads is not cartesian closed. 8 pages. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 54:3-12 (2013). E-print 1209.0414.
  15. Cyclic multicategories, multivariable adjunctions and mates. (With Nick Gurski and Emily Riehl). Journal of K-Theory, 13(2):337-396, 2014. E-print 1208.4520
  16. Iterated icons. (With Nick Gurski).  Theory and Applications of Categories, 29 (2014), 929-977E-print 1308.6495
  17. Weak infinity-categories via terminal coalgebras. (With Tom Leinster.) In Theory and Applications of Categories, 34:1073-1133, 2019. E-print 1212.5853 .
  18. Distributive laws for Lawvere Theories. In Compositionality, 2(1), 2020. E-print 1112.3076 .
  19. Weak vertical composition. (With Alexander S. Corner.) To appear in Theory and Applications of Categories. E-print 2212.11865 .
  20. The universal operad for loop spaces and generalisations. (With Todd Trimble.) To appear in Cahiers de topologie et géométrie différentielle catégoriques .

Non-specialist (for non-specialist preprints click here)


  • Weak vertical composition II: totalities. (With Alexander S. Corner.) E-print 2308.09806 .
  • Opetopic bicategories: comparison with the classical theory. Available via math.CT/0304285
  • An alternative characterisation of universal cells in opetopic n-categories, 2002. Available via math.CT/0304286
  • The theory of opetopes via Kelly-Mac Lane graphs, 2002. Available via math.CT/0304288

Ph.D. thesis, supervised by Martin Hyland

  • Higher-Dimensional Category Theory: opetopic foundations, PhD thesis, University of Cambridge, March 2002.  Available here.

Online book

  • Higher dimensional categories: an illustrated guide book.  With Aaron Lauda, 2004. Available here.

This informal guide grew out of notes we made when I was teaching Aaron about n-categories in preparation for the IMA workshop on n-categories in June 2004. We made it available before the workshop to help people, and revised it afterwards. We aimed to make it informal and friendly, with tons of pictures to help the intuition along. There are even two 3D figures to cut out and make!


This work gives an explanatory introduction to various definitions of higher-dimensional category.  The emphasis is on ideas rather than formalities; the aim is to shed light on the formalities by emphasising the intuitions that lead there.  To aid this, the tone is informal and there are copious pictures.  The definitions treated are those of Baez-Dolan, Batanin, Joyal, Leinster, May, Penon, Simpson, Street, Tamsamani and Trimble.

Slides from research talks


Conferences to 2017

From 2017 it became too much for me to keep this list up to date.


Empire Elixir, NYC 20 May, (Keynote speaker)


Imaginary, Berlin 20-23 July, (Invited speaker)

Typelevel Summit, Oslo 4 May (Keynote speaker)

Compose conference, NYC 4-5 Feb (Keynote Speaker)


CT2015,  Aveiro, Portugal, 15-19 June

CT2014, 29 June – 5 July, Cambridge

British Mathematical Colloquium and extraordinary PSSL 94, 25-28 March, University of Sheffield (an event I am co-organising with David Jordan)

CT2013 (invited speaker), 7-13 July, Macquarie University and satellite workshop 2-5 July

PSSL 93, 14-15 April 2012, Cambridge

ScotCats 4, 13 May 2011 (invited speaker), Glasgow
Category Theory, Algebra and Geometry, 26-27 May 2011 (invited speaker), Louvain-la-Neuve, Belgium
CT2011, 17-23 July UBC, Vancouver

Categories, Logic and Foundations of Physics VI, March 9 2010 Oxford, UK
PSSL 90, 24-25 April 2010, Braunschweig, Germany
CT2010, 20-26 June 2010, Genova, Italy
PSSL 91, 28-29 November 2010, Amsterdam, Netherlands

Categories, Logic and Foundations of Physics IV, January 7 2009 (invited speaker) Imperial College, London
The 88th Peripatetic Seminar on Sheaves and Logic This is a conference I organised in honour of the joint 60th birthdays of Martin Hyland and Peter Johnstone in Cambridge, April 2009.
PSSL 89, 12-13 December 2009, Louvain-la-Neuve, Belgium

Categories, Logic and Foundations of Physics I, January 9 2008 Imperial College, London
PSSL 87, March 21-23 2008 Patras, Greece
Categories, Logic and Foundations of Physics II, May 14 2008 Imperial College, London
CT2008, June 22-28 2008 Calais, France
HOCAT 08, June 30-July 5 2008 Barcelona, Spain
ATMCS 2008, July 7-11 2008 Paris 7, France

Fields Institute: Thematic Program on Geometric Applications of Homotopy Theory 2007
I co-organised the subprogram: Higher categories and their applications – Jan/Feb 2007
Workshop January 9-13
The 85th Peripatetic Seminar on Sheaves and Logic This is a conference I organised in Nice in March 2007
SECA4, June 6-9 2007 (invited speaker) Barcelona, Spain
CT2007, June 17-23 2007 Carvoeiro, Portugal

Alpine Operad workshop, March 2006 (invited speaker)
Mac Lane memorial conference, April 2006 This is a conference I co-organised in Chicago in April 2006
CT2006, July 2006 White Point, Nova Scotia

Streetfest, July 2005
Octoberfest 2005, Ottawa
Union College Conference, December 2005

The 80th Peripatetic Seminar on Sheaves and Logic This is a conference I co-organised in Cambridge in April 2004
IMA workshop on Higher Categories and their Applications, Minneapolis, June 7-18
CT2004, UBC, Vancouver, July

PSSL 79, June 28-29, Doorn, Netherlands
MFPS 19, March 19-22, Montreal, Canada
PSSL 78, Feb 15-16, Strasbourg, France

PSSL 77, Oct 5-6, Braunschweig, Germany
PSSL 76, Mar 2-3, Copenhagen, Denmark

PSSL 75, June 9-10, Stockholm, Sweden

PSSL 74, Nov 4-5, Cambridge, UK
CT2000, July 16-22, Como, Italy
PSSL 73, Apr 29-30, Braunschweig, Germany

PSSL 73, Oct 16-17, Louvain-la-Neuve, Belgium
PSSL 71, Feb 27-28, Cambridge, UK