A remark on s-torsion pairs and on the lattice of Dyck paths

TL;DR

This paper explores the relationships among classical Catalan lattices, particularly focusing on the lattice of Dyck paths, and interprets it as a lattice of subcategories inspired by s-torsion classes, enriching the understanding of their algebraic structure.

Contribution

It provides a new interpretation of the lattice of Dyck paths as a lattice of subcategories, extending the framework of s-torsion classes to this combinatorial setting.

Findings

Lattice of Dyck paths can be viewed as a lattice of subcategories.

Connections established between classical Catalan lattices and algebraic structures.

Provides a categorical perspective on Dyck path lattice.

Abstract

There are three classical lattices on the Catalan numbers: the Tamari lattice, the lattice of noncrossing partitions and the lattice of Dyck paths. The first is known to be isomorphic to the lattice of torsion classes of the path algebra of an equioriented quiver of type $A$ and the second is known to be isomorphic to its lattice of wide subcategories. Inspired by the notion of s-torsion classes of Adachi, Enomoto and Tsukamoto, in this short note we interpret the lattice of Dyck paths as a lattice of subcategories.