The representation theory of Brauer categories I: triangular categories

This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.