New structures for physics

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B. Coecke (Ed.)

New Structures for Physics

ABC


Bob Coecke
Parks Road
OX1 3QD Oxford
United Kingdom

Coecke, B. (Ed.): New Structures for Physics, Lect. Notes Phys. 813 (Springer, Berlin
Heidelberg 2011), DOI 10.1007/978-3-642-12821-9

Lecture Notes in Physics ISSN 0075-8450
ISBN 978-3-642-12820-2
DOI 10.1007/978-3-642-12821-9
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Dedicated to the many bright young theoretical
physicists that failed to escape the fate of having
to work in institutions like banks.

Preface

New? In what sense? Surely I am not the only person who, after extensively justifying why certain mathematical structures naturally arise in physics, gets questions
like: “this is all nice maths but what’s the physics?” Meanwhile I figured out what
this truly means: “I don’t see any differential equations!” Okay, this is indeed a bit
overstated. Nowadays any mathematical argument involving groups, when these are
moreover referred to as “symmetry groups”, stands a serious chance of being eligible for carrying the label “physics”. But it hasn’t always been like this. John Slater
(cf. the Slater determinant in quantum chemistry) referred to the use of group theory
in quantum physics by Weyl, Wigner et al. as der Gruppenpest, what translates as
the “plague of groups”. Even in 1975 he wrote [14]: “As soon as [my] paper became
known, it was obvious that a great many other physicists were as ‘disgusted’ as I
had been with the group-theoretical approach to the problem. As I heard later, there
were remarks made such as ‘Slater has slain the Gruppenpest’. I believe that no
other piece of work I have done was so universally popular.” Donkeys usually don’t
make the same mistake twice, . . .
. . . and, surely I am not the only person who, after extensively justifying why certain
mathematical structures naturally arise in physics, gets questions like: “this is just
the same thing in a different language!” Well, so was Copernicus’ description of
the planets as compared to Ptolemy’s. Looking back at the facts, Ptolemy’s description turned out to be more accurate, accounting even for relativistic effects. So was
abandoning the view that the Earth was the centre of the universe and that planets
move around it on hierarchies of epicycles a step backward? Of course not. Taking
the superheavy object that the sun is to be a fixed point of reference unveiled the
gravitational force as well as a critical glimpse of Newton’s laws of motion, in
terms of Galilei’s visions and Kepler’s work. Similarly, programming languages
are not just a different way of writing down 0s and 1s, but also capture the flows of
information within a computational process. Surely we wouldn’t want the whole of
mathematics to be written down entirely in terms of 0s and 1s; imagine researching
physics in terms of nothing but 0s and 1s! All this just to say that language means
structure and additional structure means additional content: using group theory is

not just about using a different language but about identifying symmetry as a key
ingredient of physics. The same goes for the structures that are discussed in this
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book: they all identify a key ingredient in physics that deserves our attention. They
moreover identify this ingredient as present in a wide range of theories, including
theories of information and computation.
The contributing research landscape. Once a subset of mathematics is accepted by
the general physics community as relevant, many physicists seem to stop making
a distinction between that piece of mathematics and the natural phenomenon this
piece of mathematics aims to describe. For this reason, there is a high entrance
fee for a mathematical structure to be awarded this privilege. But this also means
that progress in physics does go hand-in-hand with the use of new mathematical
structures. This book contains a number of such structures which recently have been
finding their way into quantum information, foundations of general relativity, quantum foundations, and quantum gravity foundations. A surprising feature of many
of these is that these structures are already heavily used in “Euro-style” computer
science, and some were even crafted for this particular purpose. In general relativity
“Scott domains” enable to reconstruct spacetime topology from the causal structure
without making any reference to smoothness [10]. Dana Scott (= male) initially
introduced these domains in the late 60s to provide semantics for the λ-calculus [13],
which plays a key role both in the foundations of mathematics and in programming
language foundations [19]. In quantum information monoidal categories [21] are
becoming more prominent, for example, for the description of particular computational models such as topological quantum computing (see [11] for a survey), and
measurement-based quantum computing (see for example [5, 6, 8]), in which the
interaction between classical and quantum data is of key importance. Earlier it was

already suggested that topological quantum field theories [2], which are functors
between certain kinds of monoidal categories, could be relevant for a theory of
quantum gravity [3, 7]. Again, these monoidal categories are of key importance
in computer science, for example, they provide semantics for linear logic [20], a
logic which is important in concurrency theory [18], the theoretical underpinning of
mobile phone networks, internet protocols, cash machines etc.
At the n-category café John Baez suggested that a less opportunistic title for this
volume would have been: “Structures you would already know about, had you been
paying proper attention”. While as title poetry this isn’t great, he is of course right,
and for more than one reason. John himself pointed to the fact that, for example,
“Category theory has been important in algebraic topology ever since its interception in 1945. It’s just taken a while for these structures to become part of the toolkit
of the average mathematical physicist.” He and Mike Stay have more examples on
page 125 of their chapter entitled “Physics, topology, logic and computation: A
Rosetta Stone” [4]. The other reason is the one I mentioned above: these structures
are already heavily used in theoretical computer science, where the play the role of
“logic of interaction” [1], “discrete (relativistic!) spacetime” [9, 12], among many
other roles.
A personal appreciation. I started my research career in the late eighties in quantum
foundations. If that didn’t already guaranty academic suicide, I moreover studied
hidden variable theories. After my PhD, in an attempt to save my career, I moved


Preface

ix

to the dying area of quantum logic within the retiring Geneva group led by Piron.
Having become aware of my mistake I moved into pure mathematics, to category
theory, an area hated by most non-category-theoretic-mathematicians, within the
retiring category theory group at McGill University. The great surprise is that after

all of this I am still standing, while many other scholars, far more brighter than I
am, lost the battle. The worst carnage in terms of academic careers surely must have
taken place in high energy physics [15, 16]. In quantum foundations the academic
death-toll is less, but this mostly has to do with the the style quantum theory is
taught in most places: “Don’t think, just do!”, resulting in not many people ending up in quantum foundations. The reason that I ended up surviving must be that
although each of |quantum foundations , |quantum logic , |category theory causes
academic disaster, |quantum categorical logic foundations proved to be some kind
of a hit in European computer science circles where, surprisingly, “foundations”
means “cool”. In those circles structural research is indeed highly appreciated, the
reason being that one simply can’t do without. Meanwhile, the membership of
our multidisciplinary group here at Oxford University Computing Laboratory [22]
has grown to 30, which besides Samson Abramsky and myself now also includes
Andreas Döring, and a zoo of DPhil (= Oxford PhD) students with backgrounds in
theoretical physics, computer science, pure mathematics, philosophy, engineering,
and even linguistics.
How did this all came about? In 2005 I organized an event called Cats, Kets and
Cloisters (CKC) at Oxford University Computing Laboratory [23]. The event aimed
to set the stage for an encounter of researchers studying mathematical structures in
computer science, quantum foundations, pure mathematicians including specialists
in logic, category theory and knot theory, and quantum informaticians. It in particular included twelve tutorial lectures by leading experts. The success of the conference what witnessed by the fact that since there was no budget to invite speakers,
these twelve leading experts all covered there own expenses. Moreover, a chain of
similar events [24–26] emerged after CKC, the most recent one being Categories,
Quanta and Concepts (CQC) at the Perimeter Institute [27].
But a low in all this was the following. When asked by several PhD students were
they could read about “this kind of stuff”, there simply wasn’t a satisfactory answer.
This is were this volume kicks in: it collates a series of tutorials that do the job.
Contributions to this volume. We start with an ABC on monoidal category theory, by
Abramsky and Tzevelekos, Baez and Stay, and Coecke and Paquette. These bulky
contributions nicely complement each other, the first one being the lecture notes of
the category theory course here at Oxford University Computing Laboratory, the

second one exemplifying how the same structures arise in very different areas, and
the third one establishing that monoidal categories have always been “out there”
in physics. The “linear” feature of these categories is then further emphasized, in
graphical realm by Selinger, and in computational realm by Haghverdi and Scott.
In particular, Selinger’s chapter is the first rock-solid comprehensive account on the
topic of graphical calculi for monoidal categories, in which he fixes several caveats
of the existing literature. Then follows a Blute-Panagaden double which applies the


x

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theory to formal distributions and formal Feynman diagrams. After that we have
a living Legend, Jim Lambek, who exposes connections between particle physics
and mathematical linguistics, an area which he pioneered in the 1950s. Next up is
domain theory, starting with a tutorial overview by Martin, followed by a detailed
account of the domain-theoretic structure on classical and quantum states by Coecke
and Martin. This is then followed by a range of structures dealing with spacetime:
first Martin and Panangaden’s application of domain theory to general relativity,
then Hiley’s use of Clifford algebras, and finally Döring and Isham’s use of topos
theory in an 180 page long mega contribution. We end with applications of monoidal
categories in quantum computational models, firstly a general account by Hines,
which is followed by Panangaden and Paquette’s survey of topological quantum
computing.
Acknowledgments We in particular thank John Baez and the attendants of the n-category café for
the “online public review process” of several chapters in this volume. Assistance in producing this
volume was provided by the EC-FP6-STREP Foundational Structures in Quantum Information and
Computation (QICS). We also acknowledge support from EPSRC Advanced Research Fellowship
EP/D072786/1 entitled The Structure of Quantum Information and its Ramifications for IT.

Oxford, England
August 2009

Bob Coecke

References
1. Abramsky, S., Gay, S.J., Nagarajan, R.: Interaction categories and the foundations of typed
concurrent programming. In: Deductive Program Design: Proceedings of the 1994 Marktoberdorf International Summer School, NATO Advanced Science Institutes Series F, pp. 35–113.
Springer, New York (1994) viii
2. Atiyah, M.: Topological quantum field theories. Publications Mathématique de l’Institut des
Hautes Etudes Scientifiques 68, 175–186 (1989) viii
3. Baez, J.C., Dolan, J.: Higher-dimensional algebra and topological quantum field theory.
J. Math. Phys. 36, 60736105. arXiv:q-alg/9503002 (1995) viii
4. Baez, J.C. Stay, M.: Physics, topology, logic and computation: A Rosetta Stone. In: Coecke,
B. (ed.) New Structures for Physics, pp. 91–166, Springer Lecture Notes in Physics, New York.
arXiv:0903.0340 (2009) viii
5. Coecke, B., Duncan, R.W.: Interacting quantum observables. In: Proceedings of the 35th
International Colloquium on Automata, Languages and Programming (ICALP), pp. 298–310,
Lecture Notes in Computer Science 5126. Springer, New York. arXiv:0906.4725 (2008) viii
6. Coecke, B., Paquette, E.O., Pavlovic, D.: Classical and quantum structuralism. In: Mackie,
I., Gay, S. (eds.) Semantic Techniques for Quantum Computation. Cambridge University
Press, Cambridge. arXiv:0904.1997 (2009) viii
7. Crane, L.: Clock and category: Is quantum gravity algebraic? J. Math. Phys. 36, 6180–6193
(1995). arXiv:gr-qc/9504038 viii
8. Duncan, R.W., Perdrix, S.: Graph states and the necessity of Euler decomposition. In: Proceedings of Computability in Europe: Mathematical Theory and Computational Practice
(CiE’09), pp. 167–177. Lecture Notes in Computer Science 5635. Springer, New York (2009).
arXiv:0902.0500 viii

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9. Lamport, L.: Time, clocks, and the ordering of events in a distributed system. Commun. ACM
21, 558–565 (1978) viii
10. Martin, K., Panangaden, P.: A domain of spacetime intervals in general relativity. Commun.
Math. Phys. 267, 563–586 (2006) viii
11. Panangaden, P., Paquette, E.O.: A categorical presentation of quantum computation with
anyons. In: Coecke, B. (ed.) New Structures for Physics, pp. 939–979. Springer Lecture Notes
in Physics, New York (2009) viii
12. Petri, C.A.: State-transition structures in physics and in computation. Int. J. Theor. Phys. 12,
979–992 (1982) viii
13. Scott, D.: Outline of a mathematical theory of computation. Technical Monograph PRG-2,
Oxford University Computing Laboratory, Oxford (1970) viii
14. Slater, J.C.: Solid-State and Molecular Theory: A Scientific Biography. Wiley, New York
(1975) vii
15. Smolin, L.: The Trouble with Physics. Houghton-Mifflin, Boston (2006) ix
16. Woit, P.: Not Even Wrong. Jonathan Cape, London (2006) ix
17. categorical logic: logic
18. concurrency: (computer science) viii
19. lambda calculus: calculus viii
20. linear logic: logic viii
21. monoidal category: category viii
22. OUCL Quantum Group. ix
23. Cats, Kets and Cloisters (CKC) conference, organized by Bob Coecke, Oxford
University Computing Laboratory, July 17–23, 2006. :8080/
FOCS/CKCinOXFORD en.html ix
24. Quantum Physics and Logic (QPL) workshop series, organized by Bob Coecke, Prakash
Panangaden and Peter Selinger, started 2008. />QPL 09.html ix

25. Categories, Logic and Foundations of Physics (CLAP) workshop series, organized by
Bob Coecke and Andreas Döring, Imperial College and Oxford, started 2008. http://
categorieslogicphysics.wikidot.com/ ix
26. Informatic Phenomena (IP) workshop series, organized by Keye Martin and Mike Mislove,
New Orleans, started 2008. on Informatic Phenomena.html ix
27. Categories, Quanta, Concepts (CQC), organized by Bob Coecke, Andreas Döring and Lucien
Hardy, Perimeter Institute, June 1–5, 2009. />Conferences/Conferences/ ix



Contents

Part I An ABC on Compositionality
1 Introduction to Categories and Categorical Logic . . . . . . . . . . . . . . . . .
S. Abramsky and N. Tzevelekos
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Some Basic Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Universality and Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 The Curry–Howard Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Monads and Comonads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Physics, Topology, Logic and Computation: A Rosetta Stone . . . . . . . .
J. Baez and M. Stay
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Analogy Between Physics and Topology . . . . . . . . . . . . . . . . . .
2.3 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Categories for the Practising Physicist . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Coecke and É.O. Paquette
3.1 Prologue: Cooking with Vegetables . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The 1D Case: New Arrows for Your Quiver . . . . . . . . . . . . . . . . . . . .
3.3 The 2D Case: Muscle Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Quantum-Like Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Classical-Like Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.6 Monoidal Functoriality, Naturality and TQFTs . . . . . . . . . . . . . . . . . 271
3.7 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Part II Manifestations of Linearity
4 A Survey of Graphical Languages for Monoidal Categories . . . . . . . . .
P. Selinger
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Autonomous Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Traced Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Products, Coproducts, and Biproducts . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Dagger Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Bicategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Beyond a Single Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Geometry of Interaction and the Dynamics of Proof Reduction:
A Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E. Haghverdi and P. Scott
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 From Monoidal Categories to *-Autonomy . . . . . . . . . . . . . . . . . . . .
5.3 Linear Logic and Categorical Proof Theory . . . . . . . . . . . . . . . . . . . .
5.4 Traced Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 What is the Geometry of Interaction? . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 GoI Interpretation of MELL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Partial Trace and Abstract Orthogonality . . . . . . . . . . . . . . . . . . . . . .
5.8 Typed GoI for MELL in *-Categories . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 1: Graphical Representation of The Trace Axioms . . . . . . . . . . .
Appendix 2: Comparing GoI Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part III More Example Applications
6 Dagger Categories and Formal Distributions . . . . . . . . . . . . . . . . . . . . . .
R. Blute and P. Panangaden
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Dagger Categories and Nuclear Ideals . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Distributions as Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.4 Categories of Formal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Vertex Groups and Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Proof Nets as Formal Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . .
R. Blute and P. Panangaden
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Functional Integrals in Quantum Field Theory . . . . . . . . . . . . . . . . . .
7.3 Linear Realizability Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 The φ-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Exponential Identities for Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 Interpreting Proof Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 Example Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

437

8 Compact Monoidal Categories from Linguistics to Physics . . . . . . . . .
J. Lambek
8.1 Compact 2-Categories and Pregroups . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Pregroups for Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Free Compact 2-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 In Search of a Compact Feynman Category . . . . . . . . . . . . . . . . . . . .
8.5 A Pogroup for QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6 From QED to the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.7 From 2-Categories to Bicategories . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8 Other Operations in Bilinear Logic . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.9 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

437
438
442
445
453
456
459
463
465
467
468
470
471
475
477
479
481
485
486
487

Part IV Informatic Geometry
9 Domain Theory and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
K. Martin
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2 The Basic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Instances of Partiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5 The Informatic Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6 Forms of Process Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7 Provocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

491
491
494
508
524
539
559
587
589


xvi

Contents

10 A Partial Order on Classical and Quantum States . . . . . . . . . . . . . . . . .
B. Coecke and K. Martin
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Classical States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

593
593
595
630
655
672
682

Part V Spatio-Temporal Geometry
11 Domain Theory and General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . .
K. Martin and P. Panangaden
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Domains, Continuous Posets and Topology . . . . . . . . . . . . . . . . . . . .
11.3 The Causal Structure of Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 Global Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Spacetime from a Discrete Causal Set . . . . . . . . . . . . . . . . . . . . . . . . .
11.6 Spacetime as a Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.7 Time and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.8 Spacetime Geometry from a Discrete Causal Set . . . . . . . . . . . . . . . .
11.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 Process, Distinction, Groupoids and Clifford Algebras:
an Alternative View of the Quantum Formalism . . . . . . . . . . . . . . . . . . .
B.J. Hiley
12.1 The Algebra of Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Some Specific Algebras of Process . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Connections with Other Mathematical Approaches . . . . . . . . . . . . . .
12.4 Some Radical New Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.5 The Conformal Clifford C2,4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6 Connections in the Clifford Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.7 Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.8 The Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.9 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 “What is a Thing?”: Topos Theory in the Foundations of Physics . . . .
A. Döring and C. Isham
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 The Conceptual Background of Our Scheme . . . . . . . . . . . . . . . . . . .
13.3 Propositional Languages and Theories of Physics . . . . . . . . . . . . . . .

687
687
689
691
692
694
695
699
701
702
703

705
705
709
715
719
731

736
740
744
749
750
753
754
762
770


Contents

xvii

13.4
13.5
13.6
13.7
13.8
13.9

778
788
809
827
837

A Higher-Order, Typed Language for Physics . . . . . . . . . . . . . . . . . .
Quantum Propositions as Sub-objects of the Spectral Presheaf . . . .

Truth Values in Topos Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The de Groote Presheaves of Physical Quantities . . . . . . . . . . . . . . .
ˆ , R and R↔ . . . . . . . . . . . . . . . . . . . . . . . . .
The Presheaves sp( A)
Extending the Quantity-Value Presheaf to an Abelian
Group Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.10 The Role of Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.11 The Category of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.12 Theories of Physics in a General Topos . . . . . . . . . . . . . . . . . . . . . . .
13.13 The General Scheme Applied to Quantum Theory . . . . . . . . . . . . . .
13.14 Characteristic Properties of Σφ , Rφ and T, w . . . . . . . . . . . . . . . . . .
13.15 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 1: Some Theorems and Constructions Used in the Main Text . .
Appendix 2: A Short Introduction to the Relevant Parts of Topos Theory .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

858
863
871
886
894
904
911
916
925
934

Part VI Geometry and Topology in Computation
14 Can a Quantum Computer Run the von Neumann Architecture? . . .
P. Hines

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 The von Neumann Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 Relevant Quantum Information Theory . . . . . . . . . . . . . . . . . . . . . . . .
14.4 Data/Code Interchangeability, and Evaluation . . . . . . . . . . . . . . . . . .
14.5 Evaluation in the One-Bit Computer . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6 Implementing Evaluation by Unitary Operations? . . . . . . . . . . . . . . .
14.7 Evaluation as Currying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.8 Basic Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.9 Categorical Closure and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . .
14.10 Abramsky and Coecke’s Categorical Foundations
for Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.11 Evaluation by Teleportation, and the vN Architecture . . . . . . . . . . . .
14.12 Naming an Unknown Arrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.13 Other Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 A Categorical Presentation of Quantum Computation with Anyons . .
P. Panangaden and É.O. Paquette
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Spin and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3 Anyons and Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

941
941
942
944
948
950
950
953

955
960
964
966
970
972
976
978
983
983
987
989


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Contents

15.4 The Algebra of Anyons: Modular Tensor Categories . . . . . . . . . . . . 992
15.5 An Example: Fibonacci Anyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014
15.6 Universal Quantum Computation with Fibonacci Anyons . . . . . . . . 1020
15.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027



Part I An ABC on
Compositionality

Chapter 1

Introduction to Categories
and Categorical Logic
S. Abramsky and N. Tzevelekos

Abstract The aim of these notes is to provide a succinct, accessible introduction
to some of the basic ideas of category theory and categorical logic. The notes are
based on a lecture course given at Oxford over the past few years. They contain
numerous exercises, and hopefully will prove useful for self-study by those seeking
a first introduction to the subject, with fairly minimal prerequisites. The coverage
is by no means comprehensive, but should provide a good basis for further study; a
guide to further reading is included.
The main prerequisite is a basic familiarity with the elements of discrete mathematics: sets, relations and functions. An Appendix contains a summary of what
we will need, and it may be useful to review this first. In addition, some prior
exposure to abstract algebra—vector spaces and linear maps, or groups and group
homomorphisms—would be helpful.

1.1 Introduction
Why study categories—what are they good for? We can offer a range of answers for
readers coming from different backgrounds:
• For mathematicians: category theory organises your previous mathematical
experience in a new and powerful way, revealing new connections and structure,
and allows you to “think bigger thoughts”.
• For computer scientists: category theory gives a precise handle on important notions such as compositionality, abstraction, representation-independence,
genericity and more. Otherwise put, it provides the fundamental mathematical
structures underpinning many key programming concepts.
S. Abramsky (B)

OUCL, University of Oxford, Oxford, UK
e-mail:
N. Tzevelekos (B)
OUCL, University of Oxford, Oxford, UK
e-mail:
Abramsky, S., Tzevelekos, N.: Introduction to Categories and Categorical Logic. Lect. Notes
Phys. 813, 3–94 (2011)
c Springer-Verlag Berlin Heidelberg 2011
DOI 10.1007/978-3-642-12821-9_1


4

S. Abramsky and N. Tzevelekos

• For logicians: category theory gives a syntax-independent view of the fundamental structures of logic, and opens up new kinds of models and interpretations.
• For philosophers: category theory opens up a fresh approach to structuralist
foundations of mathematics and science; and an alternative to the traditional
focus on set theory.
• For physicists: category theory offers new ways of formulating physical theories
in a structural form. There have inter alia been some striking recent applications
to quantum information and computation.

1.1.1 From Elements To Arrows
Category theory can be seen as a “generalised theory of functions”, where the focus
is shifted from the pointwise, set-theoretic view of functions, to an abstract view of
functions as arrows.
Let us briefly recall the arrow notation for functions between sets.1 A function f
with domain X and codomain Y is denoted by: f : X → Y .
f

Diagrammatic notation: X −→ Y .
The fundamental operation on functions is composition: if f : X → Y and g : Y →
Z , then we can define g ◦ f : X → Z by g ◦ f (x) := g( f (x)).2 Note that, in
order for the composition to be defined, the codomain of f must be the same as the
domain of g.
f

g

Diagrammatic notation: X −→ Y −→ Z .
Moreover, for each set X there is an identity function on X , which is denoted by:
id X : X −→ X

id X (x) := x .

These operations are governed by the associativity law and the unit laws. For f :
X → Y , g : Y → Z, h : Z → W:
(h ◦ g) ◦ f = h ◦ (g ◦ f ) ,

f ◦ id X = f = idY ◦ f .

Notice that these equations are formulated purely in terms of the algebraic operations on functions, without any reference to the elements of the sets X , Y , Z , W .
We will refer to any concept pertaining to functions which can be defined purely
in terms of composition and identities as arrow-theoretic. We will now take a first

1

A review of basic ideas about sets, functions and relations, and some of the notation we will be
using, is provided in Appendix A.

2 We shall use the notation “:=” for “is defined to be” throughout these notes.


1 Introduction to Categories and Categorical Logic

5

step towards learning to “think with arrows” by seeing how we can replace some
familiar definitions couched in terms of elements by arrow-theoretic equivalents;
this will lead us towards the notion of category.
We say that a function f : X −→ Y is:
injective if ∀x, x ∈ X. f (x) = f (x ) ⇒ x = x ,
surjective if ∀y ∈ Y. ∃x ∈ X. f (x) = y ,
monic
epic

if ∀g, h. f ◦ g = f ◦ h
if ∀g, h. g ◦ f = h ◦ f

⇒ g =h,
⇒ g =h.

Note that injectivity and surjectivity are formulated in terms of elements, while epic
and monic are arrow-theoretic.
Proposition 1 Let f : X → Y . Then,
1. f is injective iff f is monic.
2. f is surjective iff f is epic.
Proof We show 1. Suppose f : X → Y is injective, and that f ◦ g = f ◦ h, where
g, h : Z → X . Then, for all z ∈ Z :
f (g(z)) = f ◦ g(z) = f ◦ h(z) = f (h(z)) .

Since f is injective, this implies g(z) = h(z). Hence we have shown that
∀z ∈ Z . g(z) = h(z) ,
and so we can conclude that g = h. So f injective implies f monic.
For the converse, fix a one-element set 1 = {•}. Note that elements x ∈ X are in
1–1 correspondence with functions x¯ : 1 → X , where x(•)
¯
:= x. Moreover, if
f (x) = y then y¯ = f ◦ x¯ . Writing injectivity in these terms, it amounts to the
following.
∀x, x ∈ X. f ◦ x¯ = f ◦ x¯

⇒ x¯ = x¯

Thus we see that being injective is a special case of being monic.
Exercise 2 Show that f : X → Y is surjective iff it is epic.

1.1.2 Categories Defined
Definition 3 A category C consists of:
• A collection Ob(C) of objects. Objects are denoted by A, B, C, etc.
• A collection Ar(C) of arrows (or morphisms). Arrows are denoted by f , g, h,
etc.