On geometrically defined extensions of the Temperley-Lieb category in the Brauer category

We define an infinite chain of subcategories of the partition category by introducing the left-height ($l$) of a partition. For the Brauer case, the chain starts with the Temperley-Lieb ($l=-1$) and ends with the Brauer ($l=infty$) category. The End sets are algebras, i.e., an infinite tower thereof for each $l$, whose representation theory is studied in the paper.