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Coherence via Focusing for Symmetric Skew Monoidal Categories

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Logic, Language, Information, and Computation (WoLLIC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13038))

Abstract

The symmetric skew monoidal categories of Bourke and Lack are a weakening of Mac Lane’s symmetric monoidal categories where: (i) the three structural laws of left and right unitality and associativity are not required to be invertible, they are merely natural transformations with a specific orientation; (ii) the structural law of symmetry is a natural isomorphism involving three objects rather than two. In this paper we study the structural proof theory of symmetric skew monoidal categories, progressing the project initiated by Uustalu et al. on deductive systems for categories with skew structure. We discuss three equivalent presentations of the free symmetric skew monoidal category on a set of generating objects: a Hilbert-style categorical calculus; a cut-free sequent calculus; a focused subsystem of derivations, corresponding to a sound and complete goal-directed proof search strategy for the cut-free sequent calculus. Focusing defines an effective normalization procedure for maps in the free symmetric skew monoidal category, as such solving the coherence problem for symmetric skew monoidal categories.

This work was supported by the Estonian Research Council grant PSG659 and by the ESF funded Estonian IT Academy research measure (project 2014-2020.4.05.19-0001).

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Authors and Affiliations

  1. Tallinn University of Technology, Tallinn, Estonia

    Niccolò Veltri

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  1. Niccolò Veltri
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Correspondence to Niccolò Veltri .

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Editors and Affiliations

  1. Cornell University, New York, NY, USA

    Alexandra Silva

  2. Instituto de Matematica e Estatisrica, Universidade de São Paulo, Sao Paulo, São Paulo, Brazil

    Renata Wassermann

  3. Centro de Informática, Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil

    Ruy de Queiroz

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Veltri, N. (2021). Coherence via Focusing for Symmetric Skew Monoidal Categories. In: Silva, A., Wassermann, R., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2021. Lecture Notes in Computer Science(), vol 13038. Springer, Cham. https://doi.org/10.1007/978-3-030-88853-4_12

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  • DOI: https://doi.org/10.1007/978-3-030-88853-4_12

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