Polar Brauer categories, infinitesimal braids and Lie superalgebra representations

We define a class of monoidal categories whose morphisms are diagrams, and which are enhancements and generalisations of the Brauer category obtained by adjoining infinitesimal braids, "coupons" and poles. Properties of these categories are explored, particularly diagrammatic equations. We construct functors from certain of them to categories of representations of Lie algebras and superalgebras. Applications include a diagrammatic construction of the centre of the universal enveloping superalgebra and certain "characteristic identities", as well as an analysis of certain tensor representations. We show how classical diagram categories arising in invariant theory are special cases of our constructions, placing them in a single unified framework.