Presentations of diagram categories

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. Contents 1. Introduction 1 2. Strict k-linear monoidal categories and string diagrams 3 3. The partition category Par(t) 5 4. The planar rook category PR(t) 9 5. The rook category R(t) 12 6. The rook-Brauer category RB(t) 17 7. The Motzkin category M(t) 19 References 22 2010 Mathematics Subject Classification. 18D10.

Presentations for tensor categories

“…A presentation for the partition category is given by Comes in [17], building on deep work of Abrams [1] and Kock [54] on Frobenius algebras and cobordism categories. See also [42] for some further discussion, and sketches of proofs in some cases.…”