String Diagrams for Graded Monoidal Theories with an Application to Imprecise Probability

In this technical report 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. The Para construction on monoidal actegories is a motivating example for our framework. As a case study, we show how to axiomatize a variant of the graded category ImP, recently introduced by Liell-Cock and Staton to model imprecise probability.