Properties of Fixed Points in Axiomatic Domain Theory

Fixed points play a central role in domain theory, where, traditionally, the least-xed-point operator for continuous endofunctions on complete partial orders (cpos) is used. Recently, there has been considerable interest in developing a more general axiomatic (and order-free) account of the constructions of domain theory. Peter Freyd made an essential contribution to this programme by emphasising a novel universal property for the xed-points (up to isomorphism) of functors 5, 6]. When an appropriate functor has such a xed-point, it follows that there exists a xed-point operator acting on endomorphisms 5, 2]. Moreover, the xed-point operator is characterised by the property of uniformity, stated with respect to a subcategory of strict maps 9, 5]. We investigate the equational properties that hold between xed-point terms (terms) in such an axiomatic setting. Freyd already showed in 5] that a number of basic equational properties (such as dinaturality) hold. Here we provide a complete characterisation of all the valid equations. We work in a general categorical setting based on the authors' previous work on axiomatic domain theory 10, 4]. We show that, under mild conditions, the induced xed-point operator endows the appropriate category with a unique well-behaved parameterized xed-point operator (Theorem 3). By a general completeness result (Theorem 2), it follows that the axioms of iteration theories 1] are complete for deriving all valid xed-point equations. The general completeness result follows from an, apparently new, syntactic characterisation of the iteration theory equations as inducing a maximally consistent typically ambiguous theory (Theorem 1).