Refining algebraic data types

Our purpose is to formalize two potential refinements of single-sorted algebraic data types – subalgebras and algebras which satisfy equivalence relations – by considering their categorical interpretation. We review the usual categorical formalization of singleand multi-sorted algebraic data types as initial algebras, and the correspondence between algebraic data types and endofunctors. We introduce a relation on endofunctors which corresponds to a subtyping relation on algebraic data types, and partially extend this relation to multi-sorted algebras. Finally, we explore how this relation can help us understand single-sorted algebraic data types which satisfy equivalence relations. 1 Algebraic Data Types as Initial Algebras In programming languages with type systems, a type usually represents a family of values, such as the space of 32-bit integers: Int32 ∼= Z/232Z. Some programming languages (popular examples include Haskell [Je99] and ML [MTHM97]) have type systems which allow a programmer to define her own families of values by means of a recursion equation. For example, the data type representing inductively constructed binary trees with integer values at the leaves can be defined by the equation IntTree = Int32 ] (IntTree× IntTree). Typically, names must be assigned to the functions which allow the programmer to construct values of a data type. The diagram below illustrates this particular example: Int32 Leaf && M M M M M M M M M M IntTree× IntTree Node uukkkk kkkk kkkk kk IntTree The two functions Leaf : Int32 → IntTree and Node : IntTree × IntTree → IntTree can effectively be viewed as operators in an algebra which has no axioms. Together, they induce an isomorphism ∗[email protected] 1 [Leaf, Node] : Int32 ] (IntTree× IntTree)→ IntTree. In general, if an algebraic data type has k such constructor functions, we will denote them ω1, . . . , ωk. We will call the collection Ω = {ω1, . . . , ωk} the signature of an algebra. Now, if we assume that the collection of types in a hypothetical programming language is a category K, any recursion equation can be converted to an open form to obtain an endofunctor F : K → K. In our example, F is then defined to be F (X) = Int32 ] (X ×X) F (f) = [1Int32, f × f ], where f ∈ K(A,B) for any A,B ∈ Obj(K). Note that if there is a corresponding signature Ω, we can decompose the functor into a disjoint sum of the domains of each of the operators in Ω: F (X) = ⊎ ωi∈Ω Fωi(X), where ∀i, dom(ωi) = Fωi(X). In this case, the operators in Ω induce an arrow [ω1, . . . , ωn] : F (X)→ X. This suggests a definition for an algebra. Definition 1.1 Given an endofunctor F : K → K, an object A ∈ K is an F -algebra if there exists an arrow a : F (A)→ A in K(F (A), A) [Pie91]. We can now see that the object IntTree ∈ Obj(K) is indeed an F -algebra, since there exists an arrow [Leaf, Node] : F (IntTree) → IntTree. The collection of F -algebras forms a category, with homomorphisms between these algebras acting as arrows. Definition 1.2 Given an endofunctor F : K → K and F -algebras A and B, an arrow h ∈ K(A,B) is an F -homomorphism if the following diagram commutes [Pie91]: