Novel Uses of Category Theory in Modeling OOP

Category theory has been used extensively and successfully in modeling functional programming languages (see, e.g., [22, 23, 17, 20, 25, 15]). However, it has been used to a lesser extent in understanding and modeling object-oriented programming (OOP) languages, mainly focusing on OO languages extant during the early days of OOP research [13, 16, 18, 21]. Recently, we presented a detailed outline for using operads, from category theory, to model the iterative construction of the infinite subtyping relation in Java and other generic nominally-typed OO programming languages such as C#, C++ and Scala. Besides using operads to model the construction of the subtyping relation, we believe that there are plenty of other new uses of category-theoretic tools that can help in having better models and a better understanding of mainstream OOP languages. In this extended abstract we present outlines for four potential applications of category theory in OOP research. Namely, we first present (1) a summary of our use of operads to construct the Java subtyping relation, then we present (2) the possible use of representable functors (and Yoneda’s Lemma) in modeling and understanding generic types of generic nominally-typed OOP, followed by (3) the possible use of the equivalence of category presentations to relational database schema and of cartesian-closed categories as models of functional programming to model a structural view of OOP, and, finally, we present (4) the possible use of adjoint functors to model a particularly complex feature of Java generics, namely Java erasure. Operads and Generic OO Subtyping. Earlier this year, in [10, 11], we outlined how an operad, called JSO (for Java Subtyping Operad), can be defined to model the iterative construction of the generic subtyping relation in Java and other similar generic nominally-typed OO languages such as C# and Scala. Our model makes use of two facts: the fact that the generic subtyping relation in Java exhibits intricate self-similarity, due to the existence of wildcard types (and, accordingly, the existence of three subtyping rules for generic types),