One Setting for All: Metric, Topology, Uniformity, Approach Structure

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Setmonad T we consider (T, V)-algebras and introduce (T, V)-proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T, V)-algebras and of (T, V)-proalgebras turn out to be topological over Set. Mathematics Subject Classification: 18C20, 18B30, 54E15. Key words: V-matrix, V-promatrix, (T, V)-algebra, (T, V)-proalgebra, co-Kleisli composition, ordered set, metric space, topological space, uniform space, approach space, prometric space, protopological space, proapproach space, topological category.

• Type: Journal
• Year: 2004
• Where: ACS
• Authors: Maria Manuel Clementino, Dirk Hofmann, Walter Tholen
• Keywords: ACS 2004, ACS 2007, Metric Space, Space, Topological Space