On Higher Representation Theory via Categories of type Charge-Conserving--with--Glue

In this paper we introduce a strict monoidal subcategory of the category of matrices, suitable to address a higher representation theoretic analogue of radicals (non-semisimplicity) in ordinary representation theory. We show the extent to which this analogue has analogous representation theoretic properties. To illustrate, we apply to two key problems in the study of braid representations (strict monoidal functors from the braid category $\mathsf{B}$ to the matrix category): the classification problem; and the problem of analysing the ordinary braid group representations that braid representations generate in towers.