## Coherence in Category Theory and the Church-Rosser Property (1993)

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Citations: | 2 - 0 self |

### BibTeX

@MISC{Jay93coherencein,

author = {C. Barry Jay},

title = {Coherence in Category Theory and the Church-Rosser Property},

year = {1993}

}

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### Abstract

Szabo's derivation systems on sequent calculi with exchange and product are not Church-Rosser. Thus his coherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen's sequent calculi [9] have been applied extensively in category theory, e.g [2, 3, 4, 6, 7, 8]. Sequents correspond to morphisms of a category, and the rules of the calculus correspond to categorical structures (e.g. having an associative tensor product). Cut-elimination was then used to put bounds on the complexity of these structures, e.g. to produce exhaustive lists (perhaps with duplications) of the canonical natural transformations between given functors. For symmetric, monoidal closed categories it was shown in [12] how to decide in principle whether two such transformations are equal, while an effective, linear-time decision procedure was given in [1]. Derivation systems (reduction rules) can be used to eliminate some duplicates in the list of cut-free ...

### Citations

449 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...categories (mentioned above) and for cartesian closed categories. The former claim leads to self-contradictory assertions about the size of homsets in free categories. The latter problem is solved in =-=[5]-=- using similar methods, but avoids adopting symmetry as a primitive by exploiting the universal property of the cartesian product. I would like to thank Professor S. Mac Lane for raising this issue an... |

226 |
Closed categories
- Eilenberg, Kelly
- 1966
(Show Context)
Citation Context ...oherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen's sequent calculi [9] have been applied extensively in category theory, e.g =-=[2, 3, 4, 6, 7, 8]-=-. Sequents correspond to morphisms of a category, and the rules of the calculus correspond to categorical structures (e.g. having an associative tensor product). Cut-elimination was then used to put b... |

118 |
The Collected Papers of Gerhard Gentzen
- Szabo
- 1969
(Show Context)
Citation Context ...ith exchange and product are not Church-Rosser. Thus his coherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen's sequent calculi =-=[9]-=- have been applied extensively in category theory, e.g [2, 3, 4, 6, 7, 8]. Sequents correspond to morphisms of a category, and the rules of the calculus correspond to categorical structures (e.g. havi... |

58 |
Deductive systems and categories II. Standard constructions and closed categories. In: Category Theory, Homology Theory and their
- Lambek
- 1969
(Show Context)
Citation Context ...oherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen's sequent calculi [9] have been applied extensively in category theory, e.g =-=[2, 3, 4, 6, 7, 8]-=-. Sequents correspond to morphisms of a category, and the rules of the calculus correspond to categorical structures (e.g. having an associative tensor product). Cut-elimination was then used to put b... |

18 |
Deductive systems and categories I. Syntactic calculus and residuated categories
- Lambek
- 1968
(Show Context)
Citation Context ...oherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen's sequent calculi [9] have been applied extensively in category theory, e.g =-=[2, 3, 4, 6, 7, 8]-=-. Sequents correspond to morphisms of a category, and the rules of the calculus correspond to categorical structures (e.g. having an associative tensor product). Cut-elimination was then used to put b... |

16 |
Algebra of proofs
- Szabo
- 1978
(Show Context)
Citation Context ...linear-time decision procedure was given in [1]. Derivation systems (reduction rules) can be used to eliminate some duplicates in the list of cut-free proofs (e.g. [8]). However, in Algebra of Proofs =-=[11]-=- and its To appear in Notre Dame Journal of Formal Logic y Research supported by The Royal Society of Edinburgh/BP, and NSERC operating grant OGPIN 016. forerunner [10], Szabo claims to have produced ... |

9 |
Why commutative diagrams coincide with equivalent proofs
- Lane
- 1982
(Show Context)
Citation Context |

7 |
The structure of free closed categories
- Jay
- 1990
(Show Context)
Citation Context .... For symmetric, monoidal closed categories it was shown in [12] how to decide in principle whether two such transformations are equal, while an effective, linear-time decision procedure was given in =-=[1]-=-. Derivation systems (reduction rules) can be used to eliminate some duplicates in the list of cut-free proofs (e.g. [8]). However, in Algebra of Proofs [11] and its To appear in Notre Dame Journal of... |

3 |
Cut-elimination theorem in relevant logics
- Minc
- 1976
(Show Context)
Citation Context |

3 |
Closed categories and the theory of proofs
- Minc
- 1977
(Show Context)
Citation Context |

3 |
Coherence and non-commutative diagrams in closed categories
- Voreadou
- 1977
(Show Context)
Citation Context ...ese structures, e.g. to produce exhaustive lists (perhaps with duplications) of the canonical natural transformations between given functors. For symmetric, monoidal closed categories it was shown in =-=[12]-=- how to decide in principle whether two such transformations are equal, while an effective, linear-time decision procedure was given in [1]. Derivation systems (reduction rules) can be used to elimina... |

1 |
A categorical equivalence of proofs", Notre Dame
- Szabo
- 1974
(Show Context)
Citation Context .... However, in Algebra of Proofs [11] and its To appear in Notre Dame Journal of Formal Logic y Research supported by The Royal Society of Edinburgh/BP, and NSERC operating grant OGPIN 016. forerunner =-=[10]-=-, Szabo claims to have produced derivation systems in which all duplicates have been eliminated, so that every proof has a unique normal form. In fact, none of the systems there which include a symmet... |