# effectful category

An *effectful category* is an identity-on-objects functor $π β β$ from a monoidal category $π$ (representing the pure morphisms, or “values”) to a premonoidal category $β$ (representing the effectful morphisms, or “computations”), that strictly preserves all of the premonoidal structure and whose image is central.

An effectful category $π β β$ is strict when both its monoidal and premonoidal categories are. A *Freyd category*
(Levy, 2004) is an effectful category where the pure morphisms form a cartesian monoidal category.

Effectful categories assemble into the 2-category of effectful categories.