Categorification of the Catalan monoid

TL;DR

This paper constructs a 2-category whose algebraic structure matches the semigroup algebra of a specific monoid of transformations, providing a categorification of the Catalan monoid.

Contribution

It introduces a new finitary additive 2-category that categorifies the Catalan monoid, linking it to the Grothendieck ring and semigroup algebra.

Findings

The 2-category's Grothendieck ring is isomorphic to the semigroup algebra of the Catalan monoid.

Provides a categorification framework for the Catalan monoid.

Establishes connections between 2-categories and algebraic structures of transformation monoids.

Abstract

We construct a finitary additive 2-category whose Grothendieck ring is isomorphic to the semigroup algebra of the monoid of order-decreasing and order-preserving transformations of a finite chain.