Presentations of diagram categories

TL;DR

This paper introduces and characterizes several diagram categories, providing generators, relations, and bases for their morphism spaces, which are described using specific diagram types.

Contribution

It formally defines the planar rook, rook, rook-Brauer, and Motzkin categories with generators, relations, and bases, advancing the algebraic understanding of these diagram categories.

Findings

Morphisms have bases given by specific diagram types.

Categories are described via generators and relations.

Linear bases correspond to planar rook, rook, rook-Brauer, and Motzkin diagrams.

Abstract

We describe the planar rook category, the rook category, the rook-Brauer category, and the Motzkin category in terms of generators and relations. We show that the morphism spaces of these categories have linear bases given by planar rook diagrams, rook diagrams, rook-Brauer diagrams, and Motzkin diagrams.