Liminf convergence in Ω-categories

Abstract

We investigate and unify notions of convergence in preorders and metric spaces. The basic structures we use are categories enriched over commutative unital quantales. Such structures encompass both preorders and metric spaces, as was observed already by Lawvere (1974). In this report we define a notion of convergence on the general structures that generalizes least upper bounds of chains in preorders and limits of Cauchy sequences in metric spaces. Thereby the notions of complete partial order and complete metric space are also unified, as are the two versions of continuous functions. We use this unified setting to indicate how recursive domain equations can be treated by providing a generalization of Scott's inverse limit theorem. The theorem as well as its proof specializes to Scott's original ones for the preorder case and to America and Rutten's for the metric case. We also give a categorical analysis of our notion of convergence, comparing it to weighted limits and colimits in enriched categories.