Universal Algebra and Effectful Computation

Abstract clones serve as an algebraic presentation of the syntax of a simple type theory. From the perspective of universal algebra, they define algebraic theories like those of groups, monoids and rings. This link allows one to study the language of simple type theory from the viewpoint of universal algebra. Programming languages, however, are much more complicated than simple type theory. Many useful features like reading, writing, and exception handling involve interacting with the environment; these are called side-effects. Algebraic presentations for languages with the appropriate syntax for handling effects are given by premulticategories and effectful multicategories. We study these structures with the aim of defining a suitable notion of an algebra. To achieve this goal, we proceed in two steps. First, we define a tensor on $[\to,\category{Set}]$, and show that this tensor along with the cartesian product gives the category a duoidal structure. Secondly, we introduce the novel notion of a multicategory enriched in a duoidal category which generalize the traditional notion of a multicategory. Further, we prove that an effectful multicategory is the same as a multicategory enriched in the duoidal category $[\to,\category{Set}]$. This result places multicategories and effectful multicategories on a similar footing, and provides a mechanism for transporting concepts from the theory of multicategories (which model pure computation) to the theory of effectful multicategories (which model effectful computation). As an example of this, we generalize the definition of a 2-morphism for multicategories to the duoidally enriched case. Our equivalence result then gives a natural definition of a 2-morphism for effectful multicategories, which we then use to define the notion of an algebra.