Pseudocommutativity and Lax Idempotency for Relative Pseudomonads

Abstract: We extend the classical work of Kock on strong and commutative monads, as well as the work of Hyland and Power for 2-monads, in order to define strong and pseudocommutative relative pseudomonads. In order to achieve this, we work in the more general setting of 2-multicategories rather than monoidal 2-categories. We prove analogous implications to the classical work: that a strong relative pseudomonad is a pseudo-multifunctor, and that a pseudocommutative relative pseudomonad is a multicategorical pseudomonad. Furthermore, we extend the work of L\'opez Franco with a proof that a lax-idempotent strong relative pseudomonad is pseudocommutative. We apply the results of this paper to the example of the presheaf relative pseudomonad.