Pseudocommutativity and lax idempotency for relative pseudomonads

Andrew Slattery

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. 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 López 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.

Keywords: category theory, monad theory, presheaf

2020 MSC: Primary 18N15; Secondary 18D65, 18A05, 18M65

Theory and Applications of Categories, Vol. 39, 2023, No. 34, pp 1018-1049.

Published 2023-12-11.

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