  This paper introduces monoidal (super)categories resembling the Brauer
category. For all categories, we can construct bases of the hom-spaces using
Brauer diagrams. These categories include the Brauer category, its deformation
the BWM-category, the periplectic Brauer category, and its deformation the
periplectic $q$-Brauer category but also some new exotic categories. We show
that the BWM-category is the unique deformation of the Brauer category in this
framework, while the periplectic Brauer category has two deformations, which
are each other monoidal opposite.
