## Welcome!

This is the homepage of the workshop series *Categories, Logic and Foundations of Physics*. These workshops bring together researchers from the fields mentioned in the title and promote research on structural and conceptual aspects of fundamental physical theories, operational methodologies for general physical theories, as well as the general study of mathematical structures describing dynamics and space-time.

Please feel free to contact the workshop organizers if you have any questions, or if you want to attend one of our meetings.

- Bob Coecke, Oxford University Computing Laboratory
- Andreas Doering, Oxford University Computing Laboratory

There have been five workshops so far (9th January 2008, 14th May 2008, and 23rd/24th August 2008, 7th January 2009 and 6th August 2009). We intend to keep an informal atmosphere for the workshops, with no formal registration, and strongly encourage interaction and discussions between the participants.

## Previous event: **7th CLP Workshop, 21st September 2010, University of Birmingham**

The 7th Workshop on "Categories, Logic and Physics" will be held on Tuesday, September 21st, 2010 at the University of Birmingham. As always this is a one day event, there are no conference fees. A particular focus of this meeting will be topological aspects, but, as usual, other topics are welcome too.

**Note the changed venue — CLP is spreading out ;-)**

DATE AND LOCATION:

- Tuesday 21st of September 2010
- The University of Birmingham, School of Computer Science
- Room SPX-LT1 — This Lecture Theater 1 is in the ‘Sports and Exercise Science’ building Y14, about 5min walk from Computer Science (building Y9), see here.

REGISTRATION:

If you plan to attend the workshop, please send an email as soon as possible to the local organizers:

- <ku.ca.mahb.sc|srekciV.J.S#ku.ca.mahb.sc|srekciV.J.S> and/or
- <ku.ca.mahb.sc|resuaF.B#ku.ca.mahb.sc|resuaF.B>

so that we can make local arrangements. This is important for arranging for smooth lunch, so letting us know really helps you.

TRAVEL INFORMATION:

Birmingham is easily reached by train or plane (Birmingham International Airport). The train transfer from the airport to the University is approximately 45 minutes.

For train travel, you should ask for the station "University" when purchasing tickets. Most routes include a change at Birmingham New Street to the line with destination Longridge or Redditch.

ACCOMMODATION:

The workshop is intended to be a one day event and most participants will not need to stay overnight. If you need help with an accommodation feel free to contact the local organizers.

SCHEDULE: The program can be downloaded here.

- 10:30-11:00 Welcome Coffee/Tea
- 11:00-12:00 Martín Escardó, "
**Maybe locales are made out of points after all**" - 12:00-13:00 Christopher J. Mulvey, "
**Constructive Aspects of Gelfand Duality**" - 13:00-14:00 Lunch Break [we can go to Staff House, lunch is about 5GBP]
- 14:00-15:00 Ronnie Brown, "
**What is and what should be ‘higher dimensional group theory’?**" - 15:00-16:00 Catherine Meusburger, "
**Higher categories and observables for generalised Turaev-Viro models**" - 16:00-16:30 Coffee/Tea Break
- 16:30-17:30 Simon Willerton, "
**Two 2-traces**" (tentative) - 17:30-18:30 Cecilia Flori, "
**Topos Formulation of History Quantum Theory**" - 19:00-… Pub Session

TITLES AND ABSTRACTS: (in alphabetical order)

Ronnie Brown, Bangor: **What is and what should be ‘higher dimensional group theory’?**

The presentation will show, including some knot demos, some of the problems and intuitions which have led to this question, and how certain cubical algebraic structures with partial operations whose domains are given by geometric conditions have been found quite natural for expressing modes of higher dimensional subdivision and composition which are related to long term concerns in algebraic topology.

Martín Escardó, Birmingham: **Maybe locales are made out of points after all.**

Like topology in analysis, locale theory is about open sets, continuous functions, compact spaces, approximation and limit processes, and things like that. Both topology and locale theory start with opens. In topology, an open is made out of points, but in locale theory, a point is made out of opens. The localic view makes physical and computational sense: points are infinitely small (and carry an infinite amount of information), and hence are not directly observable, but each point is uniquely characterized by its (infinite) collection of observable properties. The opens are the observables, and locale theory takes the notion of observation as primitive,

and all other notions, including that of point, as derived. (Moreover, some perfectly good spaces in locale theory have a rich supply of opens without allowing any point at all, but this is not what I will emphasize in my talk).

Although the match of (physical or computational) reality with locale theory is arguably better than with topology, locale theory may be more mathematically demanding, or at least is certainly unfamiliar to most of us. In this talk I'll discuss how one can think of locales as if they were made out of points, like the spaces of classical analysis and geometry, trying to make them more familiar, manageable, and intuitive, without loss of rigour, so that we can reason and work with them efficiently.

Cecilia Flori, Perimeter: **Topos Formulation of History Quantum Theory**

In this talk I will describe a topos formulation of consistent histories obtained using the topos reformulation of standard quantum mechanics put forward by Döring and Isham. Such a reformulation leads to a novel type of logic with which to represent propositions. In the first part of the talk I will introduce the topos reformulation of quantum mechanics. I will then explain how such a reformulation can be extended so as to include temporally-ordered collections of propositions as opposed to single time propositions. Finally I will show how such an extension will lead to the possibility of assigning truth values to temporal propositions.

Catherine Meusburger, Hamburg: **Higher categories and observables for generalised Turaev-Viro models**

Generalised Turaev-Viro models that are formulated in terms of spherical categories play an important role in three-dimensional quantum

gravity, where they are interpreted as discrete path integrals or state sum models of quantised three-manifolds. We discuss the role and interpretation of these models in quantum gravity and comment on the problem of defining observables for these models. We show how this problem can be addressed by using higher categories and discuss the mathematical properties and the physical interpretation of the resulting observables. The talk is based on joint work with John W. Barrett.

Christopher J. Mulvey, University of Sussex: **Constructive Aspects of Gelfand Duality**

One of the important foundational aspects of recent approaches to developing quantum theories of space and time has been the existence of a constructive theory of Gelfand duality for commutative C*-algebras. In this talk, we shall outline the way in which this theory was developed, examine its application to the context of quantum physics, and consider its extension to the non-commutative case.

Simon Willerton, Sheffield: **Two 2-traces** (tentative)

Over recent years, in several areas of mathematics the notion of 'categorified trace' or '2-trace' has arisen. For instance, in higher

representation theory where groups act on linear categories there is the notion of a '2-character'; in Khovanov knot homology the Hochschild homology is viewed as a categorical trace. It transpires that there are actually two orthogonal, and sometimes dual, notions of 2-trace in common usage and I will explain how they arise and give various examples from various areas of mathematics.

We are glad to see you in September,

Local organizers

Steve Vickers + Bertfried Fauser

Workshop coordinators

Bob Coecke + Andreas Döring

## Previous event: **6th CLP Workshop, 9th March 2010, Oxford University Computing Laboratory**

The sixth workshop on "Categories, Logic and Foundations of Physics" will take place at

**Oxford Comlab on Tuesday, 9th March 2010, 12:00—18:20, Wolfson Building, Parks Road, Oxford OX1 3QD**

The location of the Comlab and visitor information can be found here.

SPEAKERS AND SCHEDULE:

The first talk will take place in Room 478.

- 12:00—12:50 Martin Hyland (Cambridge)

- 13:00—14:00 Lunch break

After lunch, we will change to Lecture Theatre A.

- 14:00—14.50 Urs Schreiber (Utrecht), "
**Gauge fields in an (oo,1)-topos**" — The familiar theory of smooth Spin(n)-principal bundles with connnection has a motivation from physics: for the quantum mechanics of a spinning point particle to make sense, the space it propagates in has to have a Spin-structure. Then the dynamics of the particle is encoded in a smooth differential refinement of the corresponding topological Spin(n)-principal bundle to a smooth bundle with connection. It has been known since work by Killingback and Witten that when this is generalized to the quantum mechanics of a spinning 1-dimensional object, the Spin-structure of the space has to lift to a String-structure, where the String-group is the universal 3-connected cover of the Spin group. Contrary to the Spin-group, the String-group cannot be refined to a (finite dimensional) Lie group. Therefore the question arises what a smooth differential refinement of a String-principal bundle would be, that encodes the dynamics of these 1-dimensional objects. It turns out that this has a nice answer not in ordinary smooth differential geometry, but in "higher" or "derived" differential geometry: String(n) naturally has the structure of a smooth 2-group — a differentiable group-stack. This allows to refine a topological String-principal bundle to a generalization of a differentiable nonabelian gerbe: a smooth principal 2-bundle. In the talk I want to indicate how the theory of smooth principal bundles with connection finds a natural generalization in such higher differential geometry, and in particular provides a good notion of connections on smooth String-principal bundles.

- 14:50—15.40 Pawel Blasiak (Krakow), "
**Graph Model of the Heisenberg-Weyl algebra**" — The Heisenberg-Weyl algebra, underlying most physical realizations of Quantum Theory, is considered from a combinatorial point of view. We construct a concrete model of the algebra in terms of graphs which endowed with intuitive concepts of composition and decomposition provide a rich bi-algebra structure. It will be shown how this encompass the Heisenberg-Weyl algebra, thereby providing a straightforward interpretation of the latter as a shadow of natural constructions on graphs. In this way, by focusing on algebraic structure of Quantum Theory we intend to draw attention to genuine combinatorial underpinning of its formalism. We will also discuss some combinatorial methods suitable for this graphical calculus.

- 15.40—16:10 Panel discussion:
**Why $n$-categories?**

- 16:10—16:40 Coffee break

- 16:40—17:30 Boris Zilber (Oxford), "
**On Model Theory, noncommutative geometry and physics**" — Studying possible relations between a mathematical structure and its description in a formal language Model Theory developed a hierarchy of a 'logical perfection'. On the very top of this hierarchy we discovered a new class of structures called Zariski geometries. A joint theorem by Hrushovski and the speaker (1993) indicated that the general Zariski geometry looks very much like an algebraic variety over an algebraically closed field, but in general is not reducible to an algebro-geometric object. Later the present speaker established that a typical Zariski geometry can be explained in terms of a possibly noncommutative 'co-ordinate' algebra. Moreover, conversely, many quantum algebras give rise to Zariski geometries and the correspondence 'Co-ordinate algebra - Zariski geometry' for a wide class of algebras is of the same type as that between commutative affine algebras and affine varieties. General quantum Zariski geometries can be approximated (in a certain model-theoretic sense) by quantum Zariski geometries at roots of unity. The latter are of a finitary type, where Dirac calculus has a well-defined meaning. We use this to give a mathematically rigorous calculation of the Feynman propagator in a few simple cases. References: On model theory, non-commutative geometry and physics (survey), author's web-page 2009

- 17:30—18:20 Bertfried Fauser (Birmingham), "
**Advanced graphical calculus — Hopf, Frobenius, Schur and some motivation from group theory**" — Graphical calculus has become a tool in quantum information theory, especially the Frobenius algebra structure for modelling the copying of classical information and rewriting rules. In my talk I will try to provide a more general picture including a Hopf algebra structure, Hopf algebra cohomology and the operation of composition. The underlying isoclasses of vector spaces will be countably infinite. The development will be motivated by findings in group theory, particularly the theory of group chraraters, which will serve as running example. If time permits I will also address conformal fields.

Subsequently, we will find a nice pub for dinner and drinks.

Please bring the workshop to the attention of others who might be interested.

We are looking forward to seeing you (again) at CLP,

best regards,

Bob Coecke (Oxford), Andreas Döring (Oxford)

## About this site - the video archive

This site went live on 11th December 2007. Feedback is very welcome. One of the most important features of this site is an extensive archive of recorded talks relevant to categories, logic and the foundations of physics, given by many different speakers at different events around the world (not just the CLP series). You can browse these talks by speaker or by event, you can download or stream the videos, and download the slides where available.

We are constantly adding new talks, so make sure to check back often!