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.
Highlights
When dealing with a specific kind of categories one of the first questions we might ask is ‘What kind of limits and colimits they have?’
We show that in case of semi-analytic monads the natural choice for these arities are the categories of regular polynomials over sup-semilattices, i.e. the algebras for the terminal semi-analytic monads
We show that the preservation of Płonka sums by a functor between categories of algebras ensures that the morphism of monads that induced it belonged to the appropriate subcategory of monads
Summary
When dealing with a specific kind of categories one of the first questions we might ask is ‘What kind of limits and colimits they have?’. Płonka sums were defined as operations on categories of algebras of regular equational theories with arites being semilattices They are related to the (strong) sup-semilattice decomposition of semigroups [8, 23]. The category of the monads in question (analytic, semi-analytic, and their generalizations) is a coreflexive subcategory in the category of all monads on Set and the Płonka sums have arities being categories of some kind of polynomials over the algebras for the terminal monad in this subcategory. For analytic monads Płonka sums work, in a sense, even better (i.e. the arities can be categories of linear polynomials over monoids) This is due to the fact that in the corresponding equational theories there is a good notion of a specific occurrence of a variable in a term. When Sα acts on the set A on the right and on the set B on the left, the set A ⊗α B is the usual tensor product of Sα -sets
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