nLab promonad

Idea

A promonad or profunctor monad is a monad in the bicategory Prof of small categories, profunctors, and natural transformations.

The data of a promonad on a category $C$ is equivalently the data of an identity-on-objects functor from $C$ (known as the collapse of the promonad).

Examples

• Every monad on a category $C$ induces a representable promonad on $C$.
• Every comonad on a category $C$ induces a corepresentable promonad on $C$.

Related pages

• procomonad
• monad
• profunctor

References

• Maren Justesen, Bikategorien af Profunktorer, Aarhus 1968 (pdf)
• Michel Thiébaud, Self-dual structure-semantics and algebraic categories, PhD thesis (1971)
• Richard J. Wood, Proarrows II, Cahiers de Topologie et Géométrie Différentielle Catégoriques 26 2 (1985) 135-168 [numdam:CTGDC_1985__26_2_135_0]
• Patrick Schultz. “Regular and exact (virtual) double categories.” arXiv:1505.00712 (2015).