## Finite dimensional vector spaces are complete for traced symmetric monoidal categories

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Venue: | in: Pillars of Computer Science: Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday, Lecture Notes in Computer Science 4800 (2008 |

Citations: | 7 - 0 self |

### BibTeX

@INPROCEEDINGS{Hasegawa_finitedimensional,

author = {Masahito Hasegawa and Martin Hofmann and Gordon Plotkin},

title = {Finite dimensional vector spaces are complete for traced symmetric monoidal categories},

booktitle = {in: Pillars of Computer Science: Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday, Lecture Notes in Computer Science 4800 (2008},

year = {},

pages = {367--385}

}

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### Abstract

Abstract. We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two different arrows in the free traced category there always exists astrongtracedfunctorintoFinVectk which distinguishes them. Therefore two arrows in the free traced category are the same if and only if they agree for all interpretations in FinVectk. 1

### Citations

945 | Categories for the working mathematician - Lane - 1971 |

244 |
Braided tensor categories
- Joyal, Street
- 1993
(Show Context)
Citation Context ...ese categories were introduced to provide a categorical structure for cyclic phenomena arising in various areas of mathematics, in particular knot theory [17]; they are (balanced) monoidal categories =-=[5]-=- enriched with a trace, a natural generalization of the traditional notion of trace in linear algebra that can be thought of as a ‘loop’ operator. In computer science, specialized versions of traced m... |

168 |
Traced monoidal categories
- Joyal, Street, et al.
- 1996
(Show Context)
Citation Context ...nd other fields, and have long been of interest to Professor Trakhtenbrot: see, e.g., [15, 9, 16, 8]. In this paper they arise in connection with Joyal, Street and Verity’s traced monoidal categories =-=[6]-=-. These categories were introduced to provide a categorical structure for cyclic phenomena arising in various areas of mathematics, in particular knot theory [17]; they are (balanced) monoidal categor... |

62 | Models of Sharing Graphs: A Categorical Semantics of Let and Letrec
- Hasegawa
- 1999
(Show Context)
Citation Context ...articular, Hyland and Hasegawa independently observed a bijective correspondence between Conway (Bekič, or dinatural diagonal) fixpoint operators [1, 11] and traces on categories with finite products =-=[2, 3]-=-. Thus, the notion of trace very neatly characterises the well-behaved fixpoint operators commonly used in computer science. More generally, traced symmetric monoidal categories equipped with the addi... |

45 | Recursion from cyclic sharing: traced monoidal categories and models of cyclic lambda-calculi
- Hasegawa
- 1997
(Show Context)
Citation Context ...articular, Hyland and Hasegawa independently observed a bijective correspondence between Conway (Bekič, or dinatural diagonal) fixpoint operators [1, 11] and traces on categories with finite products =-=[2, 3]-=-. Thus, the notion of trace very neatly characterises the well-behaved fixpoint operators commonly used in computer science. More generally, traced symmetric monoidal categories equipped with the addi... |

30 | Complete axioms for categorical fixed-point operators
- Simpson, Plotkin
- 2000
(Show Context)
Citation Context ...back operators as well as cyclic data structures. In particular, Hyland and Hasegawa independently observed a bijective correspondence between Conway (Bekič, or dinatural diagonal) fixpoint operators =-=[1,11]-=- and traces on categories with finite products [2,3]. Thus, the notion of trace very neatly characterises the well-behaved fixpoint operators commonly used in computer science. More generally, traced ... |

15 |
Network Algebra
- S¸tefǎnescu
- 2000
(Show Context)
Citation Context ...om cyclic data structures, see ibid. Inthiscontext, freely generated traced symmetric monoidal categories can be characterised as categories of cyclic networks, and so are of particular interest (see =-=[14]-=- for a related treatment). A. Avron et al. (Eds.): Trakhtenbrot/Festschrift, LNCS 4800, pp. 367–385, 2008. c○ Springer-Verlag Berlin Heidelberg 2008368 M. Hasegawa, M. Hofmann, and G. Plotkin We char... |

9 | Categorical completeness results for the simply-typed lambda-calculus, Typed Lambda Calculi and Applications
- Simpson
- 1995
(Show Context)
Citation Context ...ach differs substantially from our model-theoretic one. In the cartesian case one considers the typed λcalculus, where there is a good deal of work, starting with Friedman’s completeness theorem: see =-=[10]-=- and the references given there for further developments. The combination of higher-order structure and traces could be an interesting subject for investigation; specifically one might consider the ca... |

7 |
Nets and data flow interpreters
- Rabinovich, Trakhtenbrot
- 1989
(Show Context)
Citation Context .... Trakhtenbrot on the occasion of his 85th birthday. Cyclic networks of various kinds occur in computer science, and other fields, and have long been of interest to Professor Trakhtenbrot: see, e.g., =-=[15, 9, 16, 8]-=-. In this paper they arise in connection with Joyal, Street and Verity’s traced monoidal categories [6]. These categories were introduced to provide a categorical structure for cyclic phenomena arisin... |

6 | Trakhtenbrot, Synchronous circuits over continuous time: feedback reliability and completeness
- Pardo, Rabinovich, et al.
- 2004
(Show Context)
Citation Context .... Trakhtenbrot on the occasion of his 85th birthday. Cyclic networks of various kinds occur in computer science, and other fields, and have long been of interest to Professor Trakhtenbrot: see, e.g., =-=[15, 9, 16, 8]-=-. In this paper they arise in connection with Joyal, Street and Verity’s traced monoidal categories [6]. These categories were introduced to provide a categorical structure for cyclic phenomena arisin... |

5 |
Proof of a conjecture of S
- Soloviev
- 1997
(Show Context)
Citation Context ...njectured that the category of vector spaces over a field is complete for the symmetric monoidal closed category freely generated by a set of atoms. This was proved in a more general form by Soloviev =-=[12]-=-; his proof-theoretic approach differs substantially from our model-theoretic one. In the cartesian case one considers the typed λ-calculus, where there is a good deal of work, starting with Friedman’... |

5 |
Functorial Knot Theory
- Yetter
- 2001
(Show Context)
Citation Context ...nd Verity’s traced monoidal categories [6]. These categories were introduced to provide a categorical structure for cyclic phenomena arising in various areas of mathematics, in particular knot theory =-=[17]-=-; they are (balanced) monoidal categories [5] enriched with a trace, a natural generalization of the traditional notion of trace in linear algebra that can be thought of as a ‘loop’ operator. In compu... |

4 |
Nets and data flow interpreters
- Rabinovich, Trakhtenbrot
- 1989
(Show Context)
Citation Context .... Trakhtenbrot on the occasion of his 85th birthday. Cyclic networks of various kinds occur in computer science, and other fields, and have long been of interest to Professor Trakhtenbrot: see, e.g., =-=[15,9,16,8]-=-. In this paper they arise in connection with Joyal, Street and Verity’s traced monoidal categories [6]. These categories were introduced to provide a categorical structure for cyclic phenomena arisin... |

2 | Iteration theories. EATCS Monographs on Theoretical Computer Science - Bloom, Ésik - 1993 |

2 |
Symmetric monoidal sketches and categories of wirings
- Hyland, Power
- 2004
(Show Context)
Citation Context ...ion which may be of interest would be to look for completeness results for various classes of symmetric monoidal categories equipped with some natural combinations of (co)units and (co)diagonals; see =-=[4]-=- for a discussion of possible such combinations. A closely related research thread is that of higher-order structures. Concerning coherence problems in category theory, Mac Lane conjectured that the c... |

1 | On operators, realizable in logical nets, Doklady - tefanescu, G |

1 |
On operators, realizable in logical nets. Doklady AN
- Trakhtenbrot
- 1957
(Show Context)
Citation Context .... Trakhtenbrot on the occasion of his 85th birthday. Cyclic networks of various kinds occur in computer science, and other fields, and have long been of interest to Professor Trakhtenbrot: see, e.g., =-=[15,9,16,8]-=-. In this paper they arise in connection with Joyal, Street and Verity’s traced monoidal categories [6]. These categories were introduced to provide a categorical structure for cyclic phenomena arisin... |

1 |
On the power of compositional proofs for nets: relationships between completeness and modularity
- Trakhtenbrot
- 1997
(Show Context)
Citation Context .... Trakhtenbrot on the occasion of his 85th birthday. Cyclic networks of various kinds occur in computer science, and other fields, and have long been of interest to Professor Trakhtenbrot: see, e.g., =-=[15,9,16,8]-=-. In this paper they arise in connection with Joyal, Street and Verity’s traced monoidal categories [6]. These categories were introduced to provide a categorical structure for cyclic phenomena arisin... |