5 - Finitary Iteration in Enriched Settings

Summary

This chapter presents the limit-colimit coincidence in categories enriched either in complete partial orders or in complete metric spaces. This chapter thus works in settings where one has a theory of approximations of objects, either as joins of $\omega$-chains or as limits of Cauchy sequences, and with endofunctors preserving this structure. There are some additional requirements, and we discuss examples. In the settings which do satisfy those requirements, the initial algebra and the terminal coalgebras exist and their structures are inverses, giving what is known as a canonical fixed point (a limit-colimit coincidence). We recover some known results on this topic due to Smyth and Plotkin in the ordered setting and to America and Rutten in the metric setting. We also discuss applications to solving domain equations.