## A Sequent Calculus for Compact Closed Categories (1996)

Citations: | 5 - 0 self |

### BibTeX

@MISC{Shirahata96asequent,

author = {Masaru Shirahata},

title = {A Sequent Calculus for Compact Closed Categories},

year = {1996}

}

### OpenURL

### Abstract

In this paper, we introduce the system CMLL of sequent calculus and establish its correspondence with compact closed categories. CMLL is equivalent in provability to the system MLL of classical linear logic with the tensor and par connectives identified. We show that the system allows a fairly simple cut-elimination, and the proofs in the system have a natural interpretation in compact closed categories. However, the soundness of the cut-elimination procedure in terms of the categorical interpretation is by no means evident. We answer to this question affirmatively and establish the soundness by using the coherence result on compact closed categories by Kelly and Laplaza. 1 Introduction In this paper, we introduce the system CMLL of sequent calculus and establish its correspondence with compact closed categories. CMLL is equivalent in provability to the system MLL of classical linear logic with the tensor ffl and par O connectives identified. Compact closed categories are abundant in ...

### Citations

150 | Elements of interaction
- Milner
- 1993
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Citation Context ...ce, compact closed categories have appeared as a framework for concurrency. The interaction category by Abramsky [1] is such an example, and the connection of Milner's action calculus with reflection =-=[7, 8]-=- to compact closed categories has been investigated. In linear logic, compact closed categories have been considered at least by Barr [2] and Blute [3]. However, Barr's treatment was purely semantic a... |

107 |
Coherence for compact closed categories
- Kelly, Laplaza
- 1980
(Show Context)
Citation Context ... indeed preserves morphisms, i.e., its soundness with respect to the categorical interpretation. We prove the soundness by using the coherence result on compact closed categories by Kelly and Laplaza =-=[6]-=-. They showed that the free compact closed category FA on A is isomorphic to the category GA with the explicit description, the morphisms of which are very much reminiscent of cut-free proof nets for ... |

87 |
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- 1991
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Citation Context ... the connection of Milner's action calculus with reflection [7, 8] to compact closed categories has been investigated. In linear logic, compact closed categories have been considered at least by Barr =-=[2]-=- and Blute [3]. However, Barr's treatment was purely semantic and Blute's syntactic treatment did not fully exploit the structure of compact closed categories. To be precise, our system of sequent cal... |

44 |
A generalization of the functorial calculus
- EILENBERG, KELLY
- 1966
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Citation Context ... dimensional vector spaces over a field I , and they were of particular interest to category theorists since they posed a serious obstacle to the general (abstract) treatment of the coherence problem =-=[4, 5]-=-. In computer science, compact closed categories have appeared as a framework for concurrency. The interaction category by Abramsky [1] is such an example, and the connection of Milner's action calcul... |

28 |
logic, coherence and dinaturality
- Blute, â€˜Linear
- 1993
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Citation Context ...n of Milner's action calculus with reflection [7, 8] to compact closed categories has been investigated. In linear logic, compact closed categories have been considered at least by Barr [2] and Blute =-=[3]-=-. However, Barr's treatment was purely semantic and Blute's syntactic treatment did not fully exploit the structure of compact closed categories. To be precise, our system of sequent calculus is formu... |

22 |
An abstract approach to coherence
- KELLY
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Citation Context ... dimensional vector spaces over a field I , and they were of particular interest to category theorists since they posed a serious obstacle to the general (abstract) treatment of the coherence problem =-=[4, 5]-=-. In computer science, compact closed categories have appeared as a framework for concurrency. The interaction category by Abramsky [1] is such an example, and the connection of Milner's action calcul... |

14 |
Action Calculi V: reflexive molecular forms. Unpublished, third draf
- Milner
- 1994
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Citation Context ...ce, compact closed categories have appeared as a framework for concurrency. The interaction category by Abramsky [1] is such an example, and the connection of Milner's action calculus with reflection =-=[7, 8]-=- to compact closed categories has been investigated. In linear logic, compact closed categories have been considered at least by Barr [2] and Blute [3]. However, Barr's treatment was purely semantic a... |

4 |
On natural transformations of distinguished functors and their superpositions in certain closed categories
- Soloviev
- 1987
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Citation Context ... Step 4 amounts to removing intermediate nodes and does not change the body at all. 6 Concluding remarks The coherence problem of compact closed categories with biproduct has been studied by Soloviev =-=[9]-=-. We also considered the extensions of CMLL with finite and infinitary additives (product and coproduct). The introduction of infinitary additives was motivated by the fact that the exponential ! A ca... |

1 |
Interaction categories." unpublished lecture notes
- Abramsky
- 1993
(Show Context)
Citation Context ... to the general (abstract) treatment of the coherence problem [4, 5]. In computer science, compact closed categories have appeared as a framework for concurrency. The interaction category by Abramsky =-=[1]-=- is such an example, and the connection of Milner's action calculus with reflection [7, 8] to compact closed categories has been investigated. In linear logic, compact closed categories have been cons... |