## Abstract scalars, loops, and free traced and strongly compact closed categories (2005)

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Venue: | PROCEEDINGS OF CALCO 2005, VOLUME 3629 OF SPRINGER LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 25 - 5 self |

### BibTeX

@INPROCEEDINGS{Abramsky05abstractscalars,,

author = {Samson Abramsky},

title = {Abstract scalars, loops, and free traced and strongly compact closed categories},

booktitle = {PROCEEDINGS OF CALCO 2005, VOLUME 3629 OF SPRINGER LECTURE NOTES IN COMPUTER SCIENCE},

year = {2005},

pages = {1--31},

publisher = {Springer-Verlag}

}

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### Abstract

We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured category, culminating in the strongly compact closed case. The simple geometric and combinatorial ideas underlying these constructions are emphasized. We also discuss variations where a prescribed monoid of scalars can be ‘glued in ’ to the free construction.

### Citations

921 | Categories for the working mathematician - Lane - 1998 |

374 |
Quantum Groups
- Kassel
(Show Context)
Citation Context ...notation for tensor categories has been extensively developed with a view to applications in categorical formulations of topological invariants, quantum groups, and topological quantum field theories =-=[15]-=-. Within the purely categorical literature, a forerunner of these developments is the early work of Kelly on coherence [17, 18]; while theare also several precursors in the non-categorical literature... |

265 |
Neumann The Logic of Quantum Mechanics
- Birkhoff, Von
- 1936
(Show Context)
Citation Context ...g these lines, incorporating additive as well as multiplicative features. This kind of logic, and its connection with Quantum Mechanics, is very different to the traditional notion of ‘Quantum Logic’ =-=[8]-=-. Duncan is continuing to develop this approach in his forthcoming thesis. 1.6 Overview The further structure of the paper is as follows. In Section 2 we explore the abstract notion of scalar which ex... |

165 |
Traced monoidal categories
- Joyal, Street, et al.
- 1996
(Show Context)
Citation Context ...ill serve as a clear, accessible and useful reference. Our constructions also serve to factor the Kelly-Laplaza construction [19] of the free compact closed category through the G or Int construction =-=[14, 2]-=- of the compact closed category freely generated by a traced symmetric monoidal category, which is a central part of (the mathematically civilised version of) the so-called ‘Geometry of Interaction’ [... |

149 | A categorical semantics of quantum protocols
- Abramsky
- 2004
(Show Context)
Citation Context ...ng fashion. The technical material itself should be essentially self-contained, from the level of a basic familiarity with monoidal categories (for which see e.g. [20]). 1.1 Background In recent work =-=[4, 5]-=-, the present author and Bob Coecke have developed a categorical axiomatics for Quantum Mechanics, as a foundation for high-level approaches to quantum informatics: type systems, logics, and languages... |

115 |
Autonomous Categories
- Barr
- 1979
(Show Context)
Citation Context ...Viewing monoidal categories as bicategories with a single 0-cell, this amounts to the axiom: Every object (1-cell) has an adjoint We can also view compact closed categories as *-autonomous categories =-=[7]-=- for which ⊗ = �, and hence as ‘collapsed’ models of Linear Logic [11]. 3.1 Examples – (Rel, ×): Sets, relations, and cartesian product. Here ηX ⊆ {∗} × (X × X) and we have ηX = ɛ c X = {(∗, (x, x)) |... |

110 | 2004): Towards a quantum programming language
- Selinger
(Show Context)
Citation Context ...25]. He shows that the framework of completely positive maps actingon generalized states represented by density operators, used in his previous work on the semantics of quantum programming languages =-=[24]-=-, fits perfectly into the framework of strongly compact closed categories. 1 He also showed that a simple construction (independently found and studied in some depth by Coecke [9]), which can be carri... |

107 |
Coherence for compact closed categories
- Kelly, Laplaza
- 1980
(Show Context)
Citation Context ...e, most notably, for traced symmetric monoidal and strongly compact closed categories. We aim to give a synthetic account, including some basic cases which are well known from the existing literature =-=[20, 19]-=-. We will progressively build up structure through the following levels: 1 Selinger prefers to use the term ‘dagger compact closed category’, since the notion of adjoint which is formalized by the dag... |

100 |
Towards a geometry of interaction
- Girard
(Show Context)
Citation Context ...delineation of additive and multiplicative levels of Quantum Mechanics is one of the conceptually interesting outcomes of our categorical axiomatics. (The terminology is based on that of Linear Logic =-=[12]-=- — of which our structures can be seen as ‘collapsed models’). In terms of ordinary algebra, the multiplicative level corresponds to the multilinear-algebraic aspect of Quantum Mechanics, and the addi... |

73 | New foundations for the geometry of interaction
- Abramsky, Jagadeesan
- 1994
(Show Context)
Citation Context ...gory generated by a traced monoidal category. Thus we can recover the Kelly-Laplaza construction as the composition of these two adjunctions: Cat ✛ FTr ⊥ UTr ✲ Tr−Cat ✛ G ⊥ U ✲ CC−Cat 5 Prefigured in =-=[1]-=-, and also in some unpublished lectures of Martin Hyland [13].Adjoints compose, so FCC(C) = G ◦ FTr(C). This factorization allows us to ‘rationally reconstruct’ the Kelly-Laplaza construction. The ma... |

67 | Dagger compact closed categories and completely positive maps
- Selinger
- 2005
(Show Context)
Citation Context ...c framework admits a range of models, including of course the Hilbert space formulation of quantum mechanics. Additional evidence for the scope of the framework is provided by recent work of Selinger =-=[25]-=-. He shows that the framework of completely positive maps actingon generalized states represented by density operators, used in his previous work on the semantics of quantum programming languages [24... |

54 |
Catégories tannakiennes
- DELIGNE
- 1990
(Show Context)
Citation Context ...ch scalar s : I → I induces a natural transformation sA : A ≃ ✲ I ⊗A s ⊗ 1A ✲ I ⊗A ≃ ✲ A . 2 Susbsequently, I became aware through Martin Hyland of the mathematical literature on Tannakian categories =-=[23, 10]-=-, stemming ultimately from Grothendieck. Tannakian categories embody much stronger assumptions than ours, in particular that the categories are abelian as well as compact closed, although the idea of ... |

44 |
A generalization of the functorial calculus
- EILENBERG, KELLY
- 1966
(Show Context)
Citation Context ...pological invariants, quantum groups, and topological quantum field theories [15]. Within the purely categorical literature, a forerunner of these developments is the early work of Kelly on coherence =-=[17, 18]-=-; while theare also several precursors in the non-categorical literature, notably Penrose’s diagrammatic notation for abstract tensors [22]. Diagrammatic notation has played an important role in our ... |

23 |
Catégories Tannakiennes
- RIVANO
- 1972
(Show Context)
Citation Context ...ch scalar s : I → I induces a natural transformation sA : A ≃ ✲ I ⊗A s ⊗ 1A ✲ I ⊗A ≃ ✲ A . 2 Susbsequently, I became aware through Martin Hyland of the mathematical literature on Tannakian categories =-=[23, 10]-=-, stemming ultimately from Grothendieck. Tannakian categories embody much stronger assumptions than ours, in particular that the categories are abelian as well as compact closed, although the idea of ... |

22 |
An abstract approach to coherence
- KELLY
(Show Context)
Citation Context ...pological invariants, quantum groups, and topological quantum field theories [15]. Within the purely categorical literature, a forerunner of these developments is the early work of Kelly on coherence =-=[17, 18]-=-; while theare also several precursors in the non-categorical literature, notably Penrose’s diagrammatic notation for abstract tensors [22]. Diagrammatic notation has played an important role in our ... |

21 | A categorical quantum logic
- Abramsky, Duncan
- 2006
(Show Context)
Citation Context ...m protocols is formulated as showing the commutativity of certain diagrams; so a computational theory of the above kind is directly applicable to such verifications. In a joint paper with Ross Duncan =-=[6]-=-, we have developed a system of Categorical Quantum Logic along these lines, incorporating additive as well as multiplicative features. This kind of logic, and its connection with Quantum Mechanics, i... |

15 |
Abstract physical traces. Theory and Applications of Categories 14
- Abramsky, Coecke
- 2005
(Show Context)
Citation Context ...(f ∗ )∗ = (f∗) ∗ . 3.5 Axiomatization of Strong Compact Closure In fact, there is a more concise and elegant axiomatization of strongly compact closed categories, which takes the adjoint as primitive =-=[5]-=-. It suffices to require the following structure on a (strict) symmetric monoidal category (C, ⊗, I, τ): – A strict monoidal involutive assignment A ↦→ A ∗ on objects. – An identity-on-objects, contra... |

3 |
Retracing some paths in process algebra. CONCUR’96
- Abramsky
- 1996
(Show Context)
Citation Context ...ill serve as a clear, accessible and useful reference. Our constructions also serve to factor the Kelly-Laplaza construction [19] of the free compact closed category through the G or Int construction =-=[14, 2]-=- of the compact closed category freely generated by a traced symmetric monoidal category, which is a central part of (the mathematically civilised version of) the so-called ‘Geometry of Interaction’ [... |

3 |
Applications of negative-dimensional tensors
- Penrose
- 1969
(Show Context)
Citation Context ...se developments is the early work of Kelly on coherence [17, 18]; while theare also several precursors in the non-categorical literature, notably Penrose’s diagrammatic notation for abstract tensors =-=[22]-=-. Diagrammatic notation has played an important role in our own work with Coecke on applying our categorical axiomatics to quantum informatics, e.g. to quantum protocols [4]. For example, the essence ... |

1 |
Feedback, trace and fixed point semantics
- Katis, Sabadini, et al.
(Show Context)
Citation Context ...geometric than ours: perhaps necessarily so, since in our situation the monoidal structure, which itself has some spatial content, is added freely. Another reference is by Katis, Sabadini and Walters =-=[16]-=-. They construct a free ‘feedback category’, which is a trace minus the Yanking axiom — which is very important for the dynamics of the trace — over a monoidal category, and then formally quotient it ... |

1 |
A semantical approach to equilibria, adaptation and evolution. Unpublished manuscript
- Pavlovic
- 2004
(Show Context)
Citation Context ... (6) over a comma category of categories with involution with a specified evaluation of scalars. We note that Dusko Pavlovic has give a free construction of traced categories over monoidal categories =-=[21]-=-. His construction is elegant, but abstract and less combinatorial/geometric than ours: perhaps necessarily so, since in our situation the monoidal structure, which itself has some spatial content, is... |