The talk will survey a recent line of work, which takes a proof-theoretic approach to solving the coherence problem(s) for skew monoidal categories and related structures. I will begin by discussing the so-called Tamari order on fully-bracketed words induced by a semi-associative law (AB)C ¡= A(BC), and explain how a simple sequent calculus may account for some of its fascinating properties, such as the fact that the set of fully-bracketed words on n+1 letters forms a lattice Yn under this order, as well as a remarkable formula counting the number of intervals in Yn. Then I will recall the definition of skew monoidal categories, and explain how a more refined sequent calculus may be used to solve two related coherence problems: deciding equality of maps and enumerating homsets in free skew monoidal categories. Closely related to recent work by Bourke and Lack, this sequent calculus may be considered as a canonical construction of the free left representable skew multicategory over a set of atoms. Finally, I will briefly discuss variations of the sequent calculus capturing ‘partially skew’ monoidal categories with different normality conditions.
Based on joint work with Tarmo Uustalu and Niccol Veltri.
This video is part of Masaryk University‘s Algebra seminar.
