Opmonoidal pseudomonads

Adrian Miranda – 30 August 2023, 6 September 2023

Given an opmonoidal monad, the category of algebras inherits a monoidal structure. Similarly, braidings and symmetries lift to categories of algebras under suitable compatibility conditions. In this talk I first will review some abstract perspectives on this phenomenon [1], [2], [3]. Then we will discuss generalisations to the two-dimensional setting, now involving pseudomonads on semi-strict monoidal 2-categories, aka Gray monoids. We find that 2-categories of pseudoalgebras do inherit a monoidal structure, but one in which associativity and left and right unit laws only hold up to coherent 2-natural isomorphisms.

[1] McCrudden, P., Opmonoidal Monads, Theory and Applications of Categories, Vol.10, No.19, 2002, pp.469–485. [2] Lack, S., Limits of Lax Morphisms, Applied Categorical Structures (2005) 13: 189–203 [3] Zawadowski, M., The formal theory of monoidal monads, Journal of Pure and Applied Algebra, Volume 216, Issues 8–9, August 2012, Pages 1932–1942.