The Sequent Calculus of Skew Monoidal Categories

Tarmo Uustalu, Niccolò Veltri and Noam Zeilberger


Abstract

Szlachanyi’s skew monoidal categories are a well-motivated variation of monoidal categories in which the unitors and associator are not required to be natural isomorphisms, but merely natural transformations in a particular direction. We present a sequent calculus for skew monoidal categories, building on the recent formulation by one of the authors of a sequent calculus for the Tamari order (skew semigroup categories). In this calculus, antecedents consist of a stoup (an optional formula) followed by a context, and the connectives behave like in the standard monoidal sequent calculus except that the left rules may only be applied in stoup position. We prove that this calculus is sound and complete with respect to existence of maps in the free skew monoidal category, and moreover that it captures equality of maps once a suitable equivalence relation is imposed on derivations. We then identify a subsystem of focused derivations and establish that it contains exactly one canonical representative from each equivalence class. This coherence theorem leads directly to simple procedures for deciding equality of maps in the free skew monoidal category and for enumerating any homset without duplicates (in particular, for deciding existence of a map). Finally, and in the spirit of Lambek’s work, we describe the close connection between this proof-theoretic analysis and Bourke and Lack’s recent characterization of skew monoidal categories as left representable skew multicategories.


Agda code for the MFPS18 version of the paper

The formalization uses Agda 2.5.3 and Agda standard library 0.14.
The file Everything.agda contains the whole formalization.


Agda code for the extended version of the paper

The formalization uses Agda 2.5.3 and Agda standard library 0.14. It also type-checks using Agda 2.6.0 and Agda standard library 0.16.
The file Everything.agda contains the whole formalization.