Generalized Płonka Sums and Products

Abstract

We give an abstract categorical treatment of Płonka sums and products using lax and oplax morphisms of monads. Płonka sums with sup-semilattices as arities were originally defined as operations on algebras of regular theories. It turns out that even more general operations are available on the categories of algebras of semi-analytic monads. Their arities are the categories of the regular polynomials over any sup-semilattice, i.e. any algebra for the terminal semi-analytic monad. We also show that similar operations can be defined on the category of algebras of any analytic monad. This time we can allow the arities to be the categories of linear polynomials over any commutative monoid, i.e. any algebra for the terminal analytic monad. There are also dual operations of Płonka products. They can be defined on Kleisli categories of commutative monads.