Categories with algebraic structure—the most prominent example being monoidal categories—satisfy equational axioms only up-to coherent isomorphisms. Therefore they are pseudo algebras. We extend Lawvere’s functorial semantics to such pseudo structure: in contrast to standard strict algebras which are identified with productpreserving functors, pseudo algebras are product-preserving pseudo funct...

In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo’ (or ‘weak’) double categories – to construct a pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad onMod over itself qua pseudomonad, and show that monads in the ‘two-sided Kleisli bicategory’ ...

A class K of structures is controlled if for all cardinals λ, the relation of L∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled (see [4]).

We develop a theory of double clubs which extends Kelly’s theory of clubs to the pseudo double categories of Paré and Grandis. We then show that the club for symmetric strict monoidal categories onCat extends to a ‘double club’ on the pseudo double category Cat of ‘categories, functors, profunctors and transformations’.