Další seminář z algebry se koná 10.6.2021 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Noam Zeilberger

Skew monoidal categories and the proof-theoretic anatomy of associativity (and unitality)

Abstrakt:
Based on joint work with Tarmo Uustalu and Niccolò Veltri.

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 Y_n under this order,as well as a remarkable formula counting the number of intervals in Y_n.
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 aset of atoms.
Finally, I will briefly discuss variations of the sequent calculus capturing "partially skew" monoidal categories with different normality conditions.

References:
[1] https://arxiv.org/abs/1803.10080
[2] https://arxiv.org/abs/2003.05213
[3] https://arxiv.org/abs/2101.10487