In a monoidal category, any two morphisms can be combined in parallel using the tensor product. In a premonoidal category --- the structure that arises when morphisms carry side effects --- this unrestricted combination fails. Only central morphisms, those that commute with everything, can be tensored freely. The rest are constrained: they carry a grade, and two morphisms can only be combined when their grades are compatible. A new framework captures this situation by grading monoidal categories over partial commutative monoids --- algebraic structures where the multiplication operation is not always defined.
The construction is surprisingly unifying. Ordinary monoidal categories are the special case where the grading monoid has a single element --- every product is defined, so no restriction applies. Effectful categories, where morphisms are either pure or effectful, correspond to grading by a two-element monoid with one forbidden product. Powerset gradings model non-interfering parallel access to shared resources: two operations can combine only when they access disjoint resource sets. Interval gradings model bounded resource consumption. Each application is a different partial monoid, but the categorical machinery is the same.
The key structural result is that effectful categories form a coreflective subcategory of the full framework. This means any effectful category can be recovered from its graded generalization by a canonical projection, and conversely, any graded category can be “collapsed” into an effectful one. The two viewpoints are not merely analogous --- they are formally connected by an adjunction.
The through-claim is that partiality in combination is not an obstacle to be engineered around but a structural feature to be axiomatized. When two things cannot always be composed, the pattern of which compositions are permitted carries information as rich as the compositions themselves.