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Pseudocommutativity and lax idempotency for relative pseudomonads

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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.

http://www.tac.mta.ca/tac/volumes/39/34/39-34.pdf

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