Graded String Diagrams

We introduce string diagrams for graded symmetric monoidal categories. Our approach includes a definition of graded monoidal theory and the corresponding freely generated syntactic category. Also, we show how an axiomatic presentation for the graded theory may be modularly obtained from one for the grading theory and one for the base category, via the Para construction on monoidal actegories. As a case study, we show how to axiomatise a variant of the graded category ImP, recently introduced by Liell-Cock and Staton to model imprecise probability. This culminates in a representation, as string diagrams with grading wires, of programs with primitives for nondeterministic and probabilistic choices and conditioning.