Mario Román (“Promonads and String Diagrams for Effectful Categories”) supplies the string-diagram calculus for computational effects, where the order of side-effecting operations matters and the usual monoidal interchange law must be blocked.
A premonoidal category has a tensor that is functorial in each argument separately but need not satisfy interchange, so
The graphical calculus comes from the free runtime monoidal category, which threads a braided runtime object prints do not commute and the picture enforces sequencing. Promonads are the proarrow counterpart of monads and organise these effectful maps. The complementary notion of a morphism with a hole, a monoidal context, is produoidal rather than premonoidal.
Example. The runtime wire blocks interchange
Draw two effectful boxes each connected to the shared
-wire. A planar isotopy that would swap them must cross , which the diagram forbids, so their two orders are genuinely different string diagrams, exactly as print "a"; print "b"differs from its transpose.
Remark. Freyd, premonoidal, effectful
“Premonoidal category with a chosen centre”, “Freyd category
”, and “effectful category” are three views of the same data. The effectful presentation is the one that draws cleanly, since the runtime object makes the non-interchange visible rather than axiomatic.
Remark. Optics survive over an effectful base
Optics do not need a symmetric monoidal base. They can be built over a premonoidal / effectful category (arXiv:2305.02906), where the residual “memory” wire linking the forward and backward pass is itself threaded through effectful maps. The forward/backward optic structure therefore survives the loss of the interchange law that effectful categories model. Bidirectional data flow does not require the intervening effects to commute, so the runtime wire above and the optic residual can coexist on the same picture.