nLab procomonad

Idea

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

Examples

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

Properties

• The procomonadic functors (i.e. the forgetful functors from categories of coalgebras for procomonads) are the closure of the comonadic functors under pullback (Corollary 11 of Street 2023).
• The procomonadic functors are closed under exponentiating by a fixed category (Proposition 12 of Street 2023).

Related pages

• promonad
• comonad
• profunctor

References

• Michel Thiébaud, Self-dual structure-semantics and algebraic categories, PhD thesis (1971)

• Manfred Bernd Wischnewsky, Aspects of categorical algebra in initialstructure categories, Cahiers de topologie et géométrie différentielle 15.4 (1974): 419-444.

• Peter Johnstone and André Joyal, Continuous categories and exponentiable toposes, Journal of Pure and Applied Algebra 25.3 (1982): 255-296.

• Bob Paré, Bob Rosebrugh, and R. J. Wood, Idempotents in bicategories, Bulletin of the Australian Mathematical Society 39.3 (1989): 421-434.

• Ross Street, Wood fusion for magmal comonads, arXiv:2311.07088 (2023)