Proof Theory of Partially Normal Skew Monoidal Categories

TL;DR

This paper develops sequent calculi for partially normal skew monoidal categories, extending previous work to include invertible structural laws, and proves cut elimination and focusing properties for these calculi.

Contribution

It introduces a family of sequent calculi for partially normal skew monoidal categories, bridging skew monoidal and monoidal categories with formal proof properties.

Findings

Proved cut elimination for the new calculi

Established focusing properties for the calculi

Defined 8 weakenings of intuitionistic non-commutative linear logic

Abstract

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.