The Sequent Calculus of Skew Monoidal Categories

by   Tarmo Uustalu, et al.

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.


page 1

page 2

page 3

page 4


Deductive Systems and Coherence for Skew Prounital Closed Categories

In this paper, we develop the proof theory of skew prounital closed cate...

Proof Theory of Skew Non-Commutative MILL

Monoidal closed categories naturally model NMILL, non-commutative multip...

Proof Theory of Partially Normal Skew Monoidal Categories

The skew monoidal categories of Szlachányi are a weakening of monoidal c...

The matrix taxonomy of left exact categories

This paper is concerned with the problem of classifying left exact categ...

A sequent calculus for a semi-associative law

We introduce a sequent calculus with a simple restriction of Lambek's pr...

An Internal Language for Categories Enriched over Generalised Metric Spaces

Programs with a continuous state space or that interact with physical pr...

The syntactic side of autonomous categories enriched over generalised metric spaces

Programs with a continuous state space or that interact with physical pr...

Please sign up or login with your details

Forgot password? Click here to reset