The Sequent Calculus of Skew Monoidal Categories

Abstract

Szlachányi'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. 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. We have formalized this development in the dependently typed programming language Agda.

Proof Theory of Partially Normal Skew Monoidal Categories

“…Bourke and Lack [6] showed that skew monoidal categories are equivalent to representable skew multicategories, a weakening of representable multicategories [11]. In our previous work [24], we showed that the map constructors and equations of a (nullary-binary) representable skew multicategory are very close to and mutually definable with those of the sequent calculus for the corresponding skew monoidal category (viewed as a deductive calculus, the representable skew multicategory uses exactly the same sequent forms, but has the basic inference rules and equations chosen differently). We expect that partially normal skew monoidal categories can be analyzed in similar terms.…”

“…In [24], we showed that the free skew monoidal category Fsk(At) admits an equivalent presentation as a sequent calculus. In the latter, sequents are triples S | Γ −→ C. The antecedent is a pair of a stoup S together with a context Γ, while the succedent C is a single formula.…”

“…In previous work [24], we started investigating the proof theory of (left-)skew monoidal categories, a weakening of monoidal categories introduced by Szlachányi [22] where the unitors and associator are not required to be isomorphisms but merely transformations in a particular direction. Extending Zeilberger's [27] analysis of the Tamari order (which is the free (left-)skew semigroup category) as a sequent calculus, we introduced a sequent calculus corresponding in a precise sense to the free skew monoidal category on a set At of generating objects.…”

Deductive Systems and Coherence for Skew Prounital Closed Categories

“…We present several deductive systems giving different but equivalent presentations of the free skew prounital closed category on a set At: a categorical calculus (Hilbert-style system), a cut-free sequent calculus and a natural deduction calculus. Similarly to the skew monoidal case [35], sequents in sequent calculus have the form S | Γ −→ C and the left implication rule only applies to the formula in the stoup position. The natural deduction calculus is, under Curry-Howard correspondence, a variant of planar typed lambda-calculus [1,39].…”

“…In recent work, we have investigated the deductive systems associated to skew monoidal categories [35,37]. These are a weakening, first studied by Szlachányi [33], of monoidal categories [6,22] in which the unitors and associator are not required to be invertible, they are merely natural transformations in a particular direction.…”