Promonads and String Diagrams for Effectful Categories — Mario Román (2022; ACT 2022 (EPTCS 380, 2023)). arXiv:2205.07664 (v4, PDF).
Gives string diagrams for premonoidal and effectful categories by adding a runtime wire, and shows that effectful categories are the same as promonads — identity-on-objects functors — which places them in the double category of profunctors and gives an account of their homomorphisms and modifications.
Sources: the paper, arXiv:2205.07664v4, checked against the arXiv listing. Index: Papers.
Key definitions and results
- Definitions 2.1–2.2: binoidal and premonoidal categories
- Definition 2.4: effectful category; Freyd category
- Definition 2.5: effectful functor
- Definition 2.8, Theorem 2.14: runtime as a resource
- Definition 3.7, Theorem 3.9: promonads ↔ identity-on-objects functors
- Definitions 3.10–3.13: promonad homomorphisms and modifications