  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.
