  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.
