🤖 AI Summary
This paper establishes an algebraic semantics foundation for programming languages with computational effects (e.g., state, I/O, exceptions). To address this, it introduces the *effect multicategory* as the central semantic model and—crucially—first defines and studies multicategories enriched over a *duoidal category*, rigorously proving their equivalence. This equivalence naturally yields definitions of effect algebras and 2-morphisms. Methodologically, the work integrates duoidal category theory, pre-multicategories, tensor and Cartesian product structures on functor categories valued in sets, and enriched category theory. The main contributions are: (i) a unified algebraic semantic framework subsuming both pure and effectful computation; (ii) the first general algebraic model for effectful programming languages based on higher-order categorical structures—specifically, 2-morphisms; and (iii) a rigorous theoretical foundation enabling axiomatization, logical reasoning, and principled language design for effect systems.
📝 Abstract
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 $[ o,category{Set}]$, 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 $[ o,category{Set}]$. 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.