# Saturday & Sunday, 23 & 24th August 2008, at Oxford University

The next workshop on *Categories, Logic and Foundations of Physics* will take place at Oxford University over the weekend of August 23-24. Programs and videos of previous workshops are available on the present web page. The program for SATURDAY is:

**11:00 - 12:30 TUTORIAL I: QUANTUM FORMALISM**(Bob Coecke)**12:30 - 14:00 LUNCH****14:00 - 15:30 TUTORIAL II: CONCEPTUAL ISSUES**(Andreas Döring)**15:30 - 16:00 BREAK****16:00 - 17:00 Keith Hannabuss (Oxford, mathematics) Categories and non-associative C*-algebras in quantum field theory****17:00 - 18:00 Louis Crane (Kansas State, mathematics) Model Categories in quantum gravity**- We examine a plausible physical hypothesis which would allow us to model quantum regions as differential graded hopf algebras or as differential graded categories.

and the program for SUNDAY is:

**10:00 - 11:00 John Barrett (Nottingham, mathematics) Knots and links in braided quantum field theory**- I will explain some aspects of braided quantum field theory: how it is a generating function of knots and links, and also a little about the relation to 3d quantum gravity.

**11:00 - 11:15 BREAK****11:15 - 12:00 Simon Perdrix (Oxford, computer science) TBA****12:00 - 12:45 Mehrnoosh Sadrzadeh (Paris VII, PPS) What is the vector space content of what we say? … a categorical approach to distributed meaning****12:45 - 14:00 LUNCH****14:00 - 14:50 Steve Vickers (Birmingham, computer science) Locales via Bundles**- Locale theory, a "point-free" approach to topology, can be understood as viewing topological spaces as the spaces of models for logical theories of a particular kind. It has been found effective in a variety of fields, including pure mathematics (deriving from algebraic geometry) and computer science (notably in Abramsky's thesis). One of its most compelling virtues is that it interacts well with constructive mathematics, including the internal mathematics of toposes. In fact, it has a more satisfactory body of constructively valid results than does ordinary point-set topology. Locales have also appeared in the topos approaches to quantum mechanics of Isham and Doering at Imperial and (more explictly) of Heunen, Landsman and Spitters at Nijmegen. [par] Locales have been motivated in a variety of ways, but the path to the constructive virtues can be long and stony. I shall outline a conceptual development in terms of bundles, and going back to a technical result described by Joyal and Tierney in 1984 and known even earlier. Essentially it says that bundles over a space X are equivalent to topological spaces in the internal mathematics of "bundles of sets" (local homeomorphisms) over X. This fits an intuition that "bundle" means a space (the fibre) parametrized by points of the base space. However, there are simple examples to show that this cannot work with a point-set approach to topology - essentially because arbitrary bundles cannot be approximated closely enough by local homeomorphisms. Instead, spaces must be replaced by locales. [par] A key notion is that of "geometric" reasoning, preserved under pullback of bundles, that is more restricted than topos-valid reasoning. It is hoped that the topos-internal reasoning of the Imperial and Nijmegen groups, insofar as it is geometric, can be expressed more intuitively as fibrewise reasoning for bundles.

**14:50 - 15:40 Chris Fewster (York, mathematics) The locally covariant approach to quantum field theory in curved spacetimes**- A long-established guiding principle for quantum field theory in curved spacetimes is that it should be formulated in a local and geometrically covariant way. Recently, Brunetti, Fredenhagen and Verch have given an elegant formulation of this principle in categorical terms: a theory should be understood as a functor from a category of spacetimes to a category of star-algebras. This conceptual clarification is at the heart of a number of important results for QFT in general spacetimes, such as a proof of the spin-statistics theorem and the perturbative construction of interacting QFT. This talk reviews the locally covariant approach and some of its applications, and also describes joint work with Verch that refines and sharpens the framework.

**15:40 - 16:10 BREAK****16:10 - 17:00 Simon Willerton (Sheffield, mathematics; The Catsters) Two 2-traces**- In a monoidal category there is a notion of a trace for certain types of endomorphisms — for example the trace of an endomorphism of a finite dimensional vector space. In a monoidal bicategory there are two different notions of trace for endomorphisms which in various cases are 'dual'; I will describe these traces diagrammatically. Several examples from assorted areas will be given and I hope that the audience will come up with some more.

It was requested by the audience at the last workshop to provide tutorials, namely quantum theory for mathematicians and category theory for physicists. At this workshop we indeed provide the first, namely, tutorials on quantum theory for mathematicians. This will include detailed accounts on the many faces of the quantum formalism (von Neumann axiomatics, Dirac notation, Bloch sphere, hints to connections with monoidal categories) and on structural theorems (Gleason's theorem, Kochen-Specken theorem, Bell inequalities, GHZ-correlations, …).

As before, for reasons of logistics, could you please let us know if you intend to attend. Here is information on how to travel to Oxford, hotel information, and the location of the Computing Laboratory.

# Wednesday, 14th May 2008, at Imperial College

- The next workshop will take place on
**Wednesday, 14th May 2008, at Imperial College, from 11:00 to 18:45, in Lecture Theatre 3, Blackett Laboratory**. For directions, see here. The room is Lecture Theatre 3 on level 1 of Blackett Laboratory (north-west corner of Imperial, access from Prince Consort Road). Street level is level 2, so you have to go down one level, either using the lift or the stairs. - We are very proud to have six excellent speakers. The schedule is as follows:
**11:00-11:05: Welcome, organisational stuff****11:05 - 11:45 (+10 min. for questions and discussion): Fay Dowker, "Dynamical Logic"**- Abstract: Despite the high regard in which physicists hold General Relativity, the spacetime nature of reality has not yet fully been taken to heart in addressing the question of the interpretation of quantum mechanics. Partial progress was made by Dirac and Feynman by casting the dynamical content of quantum theory in terms of a Sum Over (spacetime) Histories (SOH). Recently it has been suggested that this SOH is part of an interpretive framework in which the rules of inference that are used to reason about physical reality are themselves subject to dynamical law. Just as General Relativity showed that geometry is not fixed and absolute, so Quantum Mechanics may be telling us that “logic” is not a fixed background but part of physics.

**11:55 - 12:35 (+10): Basil Hiley, "Towards a Quantum Geometry: Groupoids, Clifford algebras and Shadow Manifolds"**- Abstract: I will present some ideas, which although not yet formulated in terms of categories, uses the spirit of category theory especially as articulated by Lawvere. Starting from first principles, I follow Kauffman in making distinctions between two aspects of each indivisible sub-process and then order these processes to form a groupoid, which is then generalised into what I call an algebra of process. The resulting algebra is shown to be isomorphic to a hierarchy of orthogonal Clifford algebras, which, of course, include the Pauli and Dirac algebras. I then exploit the minimal left and right ideals to map from the algebra of process to a vector space inducing a light cone structure thus inverting the usual approach since here the vector space inherits its structure from the fundamental processes. Although I am working with the algebra of process, all of this is still within the conceptual framework used by Clifford himself, namely, classical physics. I then present an argument to generalise this structure to include symplectic Clifford algebras enabling us to introduce the Heisenberg group. Using the techniques applied to orthogonal CAs, I am able to show that the idempotents of this algebra map onto the points of an underlying manifold. Since the symplectic group acts in (x, p) phase space algebra, quantum mechanics demands that this structure is non-commutative and it is this feature that produces shadow manifolds. I will discuss the significance of these results.

**12:45 - 13:35: Lunch break**(please bring 5 pounds)**13:40 - 14:40 (+10): Peter Johnstone, "Topos-Theoretic Models of the Continuum"**- Abstract: We survey a number of different approaches to modelling the continuum in a topos, with particular reference to such questions as whether the real line should be viewed as a space or a locale, the ring-theoretic properties of the continuum, and whether it is sensible to require that all real functions should be continuous or even smooth.

**14:50 - 15.30 (+10): Bruce Bartlett, "Aspects of Duality in 2-Categories"**- Abstract: The notion of ‘duality’ plays an important role in quantum algebra and topological quantum field theory, as has been particularly emphasized by Baez and Dolan. One aspect of this is the idea of duals for morphisms in a 2-category, which is a generalization of the idea of rigidity in monoidal categories. I will introduce the notion of an ‘even-handed structure’ on a 2-category as a coherent means of turning right adjoints into left adjoints, and explain how this works in various examples such as fusion categories, braided monoidal categories, 2-Hilbert spaces and derived categories having a ‘trivial Serre functor’.

**15:40 - 16:30: Discussion session****16:30 - 17:00: Coffee break****17:00 - 17:40 (+10): Paul-André Melliès, "Quantum Groupoids and Logical Dualities"**- Abstract: TBA

**17:50 - 18:30 (+10): Paolo Bertozzini, "Categories of Spectral Geometries"**- Abstract: In A. Connes' non-commutative geometry, "spaces" are described "dually" as spectral triples. We provide an overview of some of the notions that we deem necessary for the development of a categorical framework in the context of spectral geometry, namely: (a) several notions of morphism of spectral geometries, (b) a spectral theory for commutative full C*-categories, (c) a tentative definition of strict-n-C*-categories, (d) spectral geometries over C*-categories. If time will allow, we will speculate on possible applications to foundational issues in quantum physics: categorical covariance, spectral quantum space-time and modular quantum gravity.

**19:00 - … : Pub session ;-)**

**There is no formal registration, but for planning reasons, please DO let us (i.e., Andreas Doering or Bob Coecke) know via email if you would like to attend**.- Since we are running this workshop on virtually no budget, please be prepared to pay 5 pounds for your lunch. Many thanks!
- We are very much looking forward to seeing you (again) at "Categories, Logic and Foundations of Physics"!

**Wednesday, January 9, 2008:** Categories, Logic and Foundations of Physics

- Workshop participant list
- Talks, videos and slides

**Official Announcement:** We hereby wish to invite you to participate in a workshop series which aims at nourishing research in the area of *Categories, Logic and Foundations of Physics*. There seems to be a substantial number of people in the Loxbridge area and beyond with interest in this field to sustain such a series. We also welcome activity from other research strands aiming to gain structural insights into foundational physical theories, for example by means of toy models, operational methodologies for general physical theories, structures for dynamics and space-time etc. Besides the workshops we plan to have an online presence in terms of a (moderated) interactive website on which open problems, news, discussions, tutorials, recordings of talks and pointers to literature will be exhibited. The scheduled date for the first meeting, which will take place at Imperial College in London, is

Wednesday, January 9, 2008.

While we do not want to stick to an a priori fixed format for the workshop, we propose the following schedule for the first meeting, to get the ball rolling:

- 12.00-13.00 Survey talk (Chris Isham, "Topos Theory in the Formulation of Theories of Physics")
- 13.00-13.30 Research talk (Chris Heunen, "A topos for algebraic quantum theory")
- 13.30-14.15 Buffet lunch session
- 14.15-15.15 Survey talk (Samson Abramsky, "Categorical Quantum Mechanics: the "Monoidal" Approach")
- 15.15-15.45 Research talk (Ross Duncan, "Classical Structures, MUBs, and Pretty Pictures")
- 15.45-16.30 General discussion session aiming at:
- stating open problems
- requests for presentation of certain topics
- plans for the future, e.g. proposals for collaboration

- 16.30-17.00 Coffee break
- 17.00-17.30 Research talk (Jamie Vicary, "A categorical framework for the quantum harmonic oscillator")
- 17.30-18.00 Research talk (Louis Crane, "Relational topology and Quantum gravity")
- 18.15-19.00 Pub session

We invite you to put forward suggestions for whom you would like to see give a (survey or) research talk, and please let us know about others who would be interested.

**Registration:** For planning reasons, please do let us know whether you will participate in the January 9 workshop! Please email us.

**Location:** The workshop will take place at the South Kensington Campus of Imperial College. For travel information, see

http://www3.imperial.ac.uk/campusinfo/southkensington

The room is Lecture Theatre 3 on level 1 of Blackett Laboratory (north-west corner of Imperial, access from Prince Consort Road). Street level is level 2, so you have to go down one level, either using the lift or the stairs.

**Lunch arrangements:** We will provide a light lunch in the form of sandwiches. Since we are running this workshop on virtually no budget, please be prepared to pay a small amount (ca. 5 pounds) to cover the costs. Thanks for your understanding!

We are looking forward to seeing you at *Categories, Logic and Foundations of Physics*.

Best regards,

Bob Coecke (Oxford) and Andreas Doering (Imperial)

- We have a facebook group dedicated to our workshops and events. The January 9th workshop is also listed as a facebook event.

**Funding:** This event enjoys some support from EPSRC ARF EP/D072786/1 and from EC-FP6-STREP QICS.