Final Dialgebras: From Categories to Allegories

• Published in: RAIRO. Informatique théorique et applications Vol. 33; no. 4-5; pp. 401 - 426
• Main Authors: Backhouse, Roland, Hoogendijk, Paul
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Paris EDP Sciences 01.07.1999
• Subjects: 68N05; allegory theory; Applied sciences; category theory; co-inductive type; Computer science; control theory; systems; Exact sciences and technology; generic programming; inductive type; polytypic programming; Programming theory; relation algebra; Theoretical computing; theory of datatypes; Relation algebra; Datatype theory; Induction; Category theory; Allegory theory; Generic programming; Programming theory
• ISSN: 0988-3754; 1290-385X
• DOI: 10.1051/ita:1999126

The study of inductive and coinductive types (like finite lists and streams, respectively) is usually conducted within the framework of category theory, which to all intents and purposes is a theory of sets and functions between sets. Allegory theory, an extension of category theory due to Freyd, is better suited to modelling relations between sets as opposed to functions between sets. The question thus arises of how to extend the standard categorical results on the existence of final objects in categories (for example, coalgebras and products) to their existence in allegories. The motivation is to streamline current work on generic programming, in which the use of a relational theory rather than a functional theory has proved to be desirable. In this paper, we define the notion of a relational final dialgebra and prove, for an important class of dialgebras, that a relational final dialgebra exists in an allegory if and only if a final dialgebra exists in the underlying category of maps. Instances subsumed by the class we consider include coalgebras and products. An important lemma expresses bisimulations in allegorical terms and proves this equivalent to Aczel and Mendler's categorical definition.