Coproducts of Monads on Set

TL;DR

This paper characterizes coproducts of monads on Set using initial algebra formulas, establishing conditions for their existence and properties, with implications for computational effects and algebraic structures.

Contribution

It provides a new formula for coproducts of consistent monads on Set and characterizes when such coproducts exist, including their injectivity properties.

Findings

Coproducts of consistent monads are characterized by initial algebra formulas.

Coproduct embeddings of monads are injective.

Two monads have a coproduct iff they share arbitrarily large fixpoints or one is an exception monad.

Abstract

Coproducts of monads on Set have arisen in both the study of computational effects and universal algebra. We describe coproducts of consistent monads on Set by an initial algebra formula, and prove also the converse: if the coproduct exists, so do the required initial algebras. That formula was, in the case of ideal monads, also used by Ghani and Uustalu. We deduce that coproduct embeddings of consistent monads are injective; and that a coproduct of injective monad morphisms is injective. Two consistent monads have a coproduct iff either they have arbitrarily large common fixpoints, or one is an exception monad, possibly modified to preserve the empty set. Hence a consistent monad has a coproduct with every monad iff it is an exception monad, possibly modified to preserve the empty set. We also show other fixpoint results, including that a functor (not constant on nonempty sets) is finitary iff every sufficiently large cardinal is a fixpoint.