2021/01/19 by Tarmo Uustalu, Niccolò Veltrì, Niccolò Veltri +1 · 8 citations
Computer Science · Mathematics · #Algebra over a field #Algebraic structures and combinatorial models #Closed monoidal category #Computer science #Discrete mathematics #Enriched category #Higher category theory #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems #Mathematics #Monoidal category #Pure mathematics #Sequent #Sequent calculus #Skew #Symmetric monoidal category #cs.LO #math.CT
paper · pdf · doi:10.4204/eptcs.333.16
published in Electronic Proceedings in Theoretical Computer Science 333, 230-246 (Open Publishing Association) · In Proceedings ACT 2020, arXiv:2101.07888
openalex publication_date 2021/01/19 · arxiv created 2021/01/26 · arxiv updated 2021/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The skew monoidal categories of Szlach'anyi are a weakening of monoidal categories where the three structural laws of left and right unitality and associativity are not required to be isomorphisms but merely transformations in a particular direction. In previous work, we showed that the free skew monoidal category on a set of generating objects can be concretely presented as a sequent calculus. This calculus enjoys cut elimination and admits focusing, i.e. a subsystem of canonical derivations, which solves the coherence problem for skew monoidal categories. In this paper, we develop sequent calculi for partially normal skew monoidal categories, which are skew monoidal categories with one or more structural laws invertible. Each normality condition leads to additional inference rules and equations on them. We prove cut elimination and we show that the calculi admit focusing. The result is a family of sequent calculi between those of skew monoidal categories and (fully normal) monoidal categories. On the level of derivability, these define 8 weakenings of the (unit,tensor) fragment of intuitionistic non-commutative linear logic.