Dependence of results on the choice of grading for ImP

Determine the extent to which the main results obtained for the graded Markov category ImP depend on choosing the grading category to be the semicartesian monoidal subcategory of FinStoch consisting of natural numbers and surjective stochastic matrices, as opposed to alternative grading categories such as finite sets with surjective functions or nominal gradings with objects of the form 2^A and surjections 2^B → 2^A induced by injections A → B.

Background

ImP is defined by grading the category of finite stochastic matrices (FinStoch) using the subcategory of natural numbers and surjective stochastic matrices, which encodes named Knightian choices and their renamings via graded structure.

The authors note that other plausible grading categories could be used, including deterministic surjective functions on finite sets or surjections between product sets 2^B → 2^A induced by name injections. Whether the paper’s core conclusions (e.g., compositionality, op-lax interpretation to convex powersets, and maximality) depend critically on this particular grading remains unresolved.