Complete Axioms for Categorical Fixed-Point Operators

Alex Simpson, Gordon Plotkin

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We give an axiomatic treatment of fixed-point operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterization of the free iteration theory.We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness). Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixed-point operator is necessarily an iteration operator.
Original languageEnglish
Title of host publicationProceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
Place of PublicationWashington, DC, USA
PublisherInstitute of Electrical and Electronics Engineers (IEEE)
Number of pages12
Publication statusPublished - 2000


  • Categorical models, domain theory, fixed points, iteration theories


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