Presentations of pseudodistributive laws
Charles Walker
TL;DR
This paper addresses the problem of presenting pseudodistributive laws between pseudomonads in a dimension-two categorical setting. It introduces five equivalent presentations—pseudomonoidal, Kleisli-decagon, pseudoalgebra, no-iteration, and warping—and proves their equivalence to the Marmolejo framework via compatible extensions to the Kleisli bicategory, while showing that three of the original coherence axioms are redundant. The approach leverages extensive and no-iteration forms, pasting operators, and coherence results like MacLane–Paré to simplify and relate the various formulations. The results yield a streamlined, flexible toolkit for pseudodistributive laws, with potential applications to relative pseudomonads and coherence in higher-dimensional settings, and highlight practical reductions in data needed to define such laws. Overall, the work clarifies the landscape of presentations and provides a robust equivalence theory that can simplify future constructions and reasoning in higher category theory.$
Abstract
By considering the situation in which the involved pseudomonads are presented in no-iteration form, we deduce a number of alternative presentations of pseudodistributive laws including a 'decagon' form, a pseudoalgebra form, a no-iteration form, and a warping form. As an application, we show that five coherence axioms suffice in the usual monoidal definition of a pseudodistributive law.