We study the existence of universal measuring comonoids P(A,B) for a pair of monoids A, B in a braided monoidal closed category, and the associated enrichment of the category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if A is a bimonoid and B is a commutative monoid, then P(A,B) is a bimonoid; in addition, if A is a cocommutative Hopf monoid then P(A,B) always is Hopf. If A is a Hopf monoid, not necessarily cocommutative, then P(A,B) is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative description of the dualizable P(A,B)‐comodules and use the theory of Hopf (co)monads. We explore the examples of universal measuring comonoids in vector spaces and graded spaces.

Abstract. This paper introduces lax orthogonal algebraic weak factorisation systems on 2-categories and describes a method of constructing them. This method rests in the notion of simple 2-monad, that is a generalisation of the simple reflections studied by Cassidy, Hébert and Kelly. Each simple 2-monad on a finitely complete 2-category gives rise to a lax orthogonal algebraic weak factorisation system, and an example of a simple 2-monad is given by completion under a class of colimits. The notions of kz lifting operation, lax natural lifting operation and lax orthogonality between morphisms are studied.

We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories enriched over a normal duoidal category; using this, we re-find notions such as the commutativity of a finitary algebraic theory or a strong monad, the commuting tensor product of two theories, and the Boardman-Vogt tensor product of symmetric operads.Comment: 48 pages; final journal versio

We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of n-monoidal categories and discuss several naturally occurring examples in which n ≤ 3.