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2004

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

10 years 1 months ago
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.
Maria Manuel Clementino, Dirk Hofmann, Walter Thol
Added 16 Dec 2010
Updated 16 Dec 2010
Type Journal
Year 2004
Where ACS
Authors Maria Manuel Clementino, Dirk Hofmann, Walter Tholen
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