Monoidal categories graded by partial commutative monoids

Motivation from the study of concurrency in computer science leads to our notion of monoidal category graded by a partial commutative monoid: morphisms are graded by elements of a partial monoid, and their monoidal product is defined only when the sum of grades is defined.
The definition of this notion involves a number of moving parts, but they can be seen to emerge from natural considerations involving promonoidal categories, convolution, and enrichment. The particular instances of these concepts are relatively simple, and so in turn provide a gentle introduction to them, which is one of the purposes of this talk.
This talk concerns work in progress with Chad Nester and Mario Román.