String Diagrams for Premonoidal Categories

Premonoidal categories are monoidal categories without the interchange law; effectful categories are premonoidal categories with a chosen monoidal subcategory of interchanging morphisms. In the same sense that string diagrams, pioneered by Joyal and Street, are an internal language for monoidal categories, we show that string diagrams with an added 'runtime wire', pioneered by Alan Jeffrey, are an internal language for effectful categories and can be used as string diagrams for effectful, premonoidal and Freyd categories.
Comment: Extended version of the first chapters of 'Promonads and String Diagrams for Effectful Categories' from the first-named author, arXiv:2205.07664, presented at Applied Category Theory 2022