Category Theory as Coherently Constructive Lattice Theory: an Illustration

Dijkstra and Scholten have formulated a theorem stating that all disjunctivity properties of a predicate transformer are preserved by the construction of least preex points. An alternative proof of their theorem is presented based on two fundamental xed point theorems, the abstraction theorem and the fusion theorem, and the fact that suprema in a lattice are deened by a Galois connection. The abstraction theorem seems to be new; the fusion theorem is known but its importance does not seem to be fully recognised. The abstraction theorem, the fusion theorem, and Dijkstra and Scholten's theorem are then generalised to the context of category theory and shown to be valid. None of the theorems in this context seems to be known, although speciic instances of Dijkstra and Scholten's theorem are known. The main point of the paper is to discuss the process of drawing inspiration from lattice theory to formulate theorems in category theory ((rst advocated by Lambek in 1968). We advance the view that, in order to contribute to the development of programming methodology, category theory may be prootably regarded as \construc-tive" lattice theory in which the added value lies in establishing that the constructions are \coherent". This paper was specially prepared for presentation at the meeting of IFIP Working Group 2.3 (Programming Methodology), June 1994. Knowledge of (elementary) lattice theory is assumed. Knowledge of category theory is not.