## The Uniformity Principle on Traced Monoidal Categories (2003)

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Venue: | In Proceedings of CTCS’02, volume 69 of ENTCS |

Citations: | 8 - 0 self |

### BibTeX

@INPROCEEDINGS{Hasegawa03theuniformity,

author = {Masahito Hasegawa},

title = {The Uniformity Principle on Traced Monoidal Categories},

booktitle = {In Proceedings of CTCS’02, volume 69 of ENTCS},

year = {2003},

pages = {1014},

publisher = {Elsevier}

}

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### Abstract

The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin's principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones. Some classical examples like the Scott induction principle are shown to be instances of these constructions. We also characterize some specific cases of our constructions as suitable enriched limits. 1

### Citations

392 | Basic Concepts of Enriched Category Theory
- Kelly
- 1982
(Show Context)
Citation Context ...ple, it is tempting to characterize C # as a Graph-cotensor of the graph ( # ) and the traced monoidal category C (see Appendix D for the notion of cotensors; for general enriched category theory see =-=[10,4]-=-). Thus we would like to claim TreMon(B, C # ) # [( # ), TreMon(B, C)] where TreMon is the Graph-enriched category of traced monoidal categories and strict traced functors as described above. Alas, th... |

169 |
Traced monoidal categories
- Joyal, Street, et al.
- 1996
(Show Context)
Citation Context ...f these constructions. We also characterize some specific cases of our constructions as suitable enriched limits. 1 Introduction Traced monoidal categories were introduced by Joyal, Street and Verity =-=[9]-=- as the categorical structure for cyclic phenomena arising from various areas of mathematics. They are (balanced) monoidal categories enriched with a trace, which is a natural generalization of the tr... |

157 | The geometry of tensor calculus - Joyal, Street - 1991 |

150 |
Semantics of Programming Languages
- Gunter
- 1992
(Show Context)
Citation Context ...ures the well-behaved fixpoint operators commonly used in computer science. An extra bonus of this trace-fixpoint correspondence is that the uniformity principle (à la Plotkin) on a fixpoint operator =-=[20, 6, 23]-=- precisely amounts to a uniformity principle on the corresponding trace, as introduced in the previous work [8, 9] (see also historical remarks at the end of Sec. 3). This uniformity principle is gene... |

104 |
Geometry of Interaction I: interpretation of Systef F
- Girard
- 1988
(Show Context)
Citation Context ...fully embeds (see Appendix B for a summary of the construction). Its applications to computer science are discussed and studied, especially with the relation to Girard’s Geometry of Interaction (GoI) =-=[5]-=- which essentially amounts to implement bidirectional (or interactive) communications via feedback operators, in the literature [1, 13, 7] — in this context, the Int construction is also known as the ... |

80 |
Handbook of categorical algebra 2. Categories and structures
- Borceux
- 1994
(Show Context)
Citation Context ...ple, it is tempting to characterize C # as a Graph-cotensor of the graph ( # ) and the traced monoidal category C (see Appendix D for the notion of cotensors; for general enriched category theory see =-=[10,4]-=-). Thus we would like to claim TreMon(B, C # ) # [( # ), TreMon(B, C)] where TreMon is the Graph-enriched category of traced monoidal categories and strict traced functors as described above. Alas, th... |

67 | Retracing some paths in process algebra, in
- Abramsky
- 1996
(Show Context)
Citation Context ...oidal category C into a compact closed category Int C to which C fully faithfully embeds (see Appendix E for a summary of the construction); its applications to computer science are studied, e.g., in =-=[1,5]-=-. It is natural to ask how the uniformity principles in C and in Int C are related. Unfortunately, the situation seems less clear than we first guess, and in this paper we can give only some elementar... |

64 |
Iteration Theories
- Bloom, ´Esik
- 1993
(Show Context)
Citation Context ... well as cyclic data structures. In the middle of 90's, Martin Hyland and the author independently observed a bijective correspondence between Conway (Bekic, or dinatural diagonal) fixpoint operators =-=[3,12]-=- and traces on categories with finite products [6,7]. Thus, in this setting, the notion of trace precisely captures the well-behaved fixpoint operators commonly used in computer science. An extra bonu... |

63 | Models of Sharing Graphs: A Categorical Semantics of let and letrec
- Hasegawa
- 1997
(Show Context)
Citation Context ...s, Martin Hyland and the author independently observed a bijective correspondence between Conway (Bekic, or dinatural diagonal) fixpoint operators [3,12] and traces on categories with finite products =-=[6,7]-=-. Thus, in this setting, the notion of trace precisely captures the well-behaved fixpoint operators commonly used in computer science. An extra bonus of this trace-fixpoint correspondence is that the ... |

45 | Recursion from cyclic sharing: traced monoidal categories and models of cyclic lambda-calculi
- Hasegawa
- 1997
(Show Context)
Citation Context ...s, Martin Hyland and the author independently observed a bijective correspondence between Conway (Bekic, or dinatural diagonal) fixpoint operators [3,12] and traces on categories with finite products =-=[6,7]-=-. Thus, in this setting, the notion of trace precisely captures the well-behaved fixpoint operators commonly used in computer science. An extra bonus of this trace-fixpoint correspondence is that the ... |

27 | A relational model of non-deterministic dataflow
- Hildebrandt, Panangaden, et al.
- 2004
(Show Context)
Citation Context ...pecially with the relation to Girard’s Geometry of Interaction (GoI) [5] which essentially amounts to implement bidirectional (or interactive) communications via feedback operators, in the literature =-=[1, 13, 7]-=- — in this context, the Int construction is also known as the GoI construction. It is interesting to see how the uniformity principles in C and in Int C are related. However, the situation seems less ... |

18 |
A categorical approach to linear logic, geometry of proofs and full completeness
- Haghverdi
- 2000
(Show Context)
Citation Context ...oidal category C into a compact closed category Int C to which C fully faithfully embeds (see Appendix E for a summary of the construction); its applications to computer science are studied, e.g., in =-=[1,5]-=-. It is natural to ask how the uniformity principles in C and in Int C are related. Unfortunately, the situation seems less clear than we first guess, and in this paper we can give only some elementar... |

5 |
Partially additive categories and flow-diagram semantics
- Arbib, Manes
- 1980
(Show Context)
Citation Context ...ork algebras [13] precisely corresponds to our uniformity principle, where strict arrows are called "functorial arrows" (following the terminology by Arbib and Manes for partially additive c=-=ategories [2]-=-). See ibid. for bibliographic remarks and also several examples. 3 Hasegawa 3 Basic Facts As a warming up, let us see a few basic (but fundamental) properties which strict maps in a traced monoidal c... |

5 | Paracategories I: internal paracategories and saturated partial algebras
- Hermida, Mateus
- 2003
(Show Context)
Citation Context ...w we can characterize the original constructions (without using specified classes of strict maps). Another direction which might be worth looking at is to replace the graphs by Freyd’s paracategories =-=[11, 12]-=- (roughly: categories in which composition is a partial operator), since strict maps form not just a graph but a paracategory, and one may expect “monoidal paracategories” would give a better account ... |

1 |
Categorical structure of asynchronomy
- Selinger
(Show Context)
Citation Context ...ity principle is general enough to make sense for arbitrary traced monoidal categories. An application of this concept is found in Selinger's work on categorical models of asynchronous communications =-=[11]-=-. c #2003 Published by Elsevier Science B. V. Hasegawa In the present paper we demonstrate that this uniformity principle on traced monoidal categories can be used for constructing new traced monoidal... |

1 |
II: adjunctions, fibrations and examples from probabilistic automata theory, Theoret
- Paracategories
(Show Context)
Citation Context ...w we can characterize the original constructions (without using specified classes of strict maps). Another direction which might be worth looking at is to replace the graphs by Freyd’s paracategories =-=[11, 12]-=- (roughly: categories in which composition is a partial operator), since strict maps form not just a graph but a paracategory, and one may expect “monoidal paracategories” would give a better account ... |