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Applied Categorical Structures

, Volume 12, Issue 2, pp 127–154 | Cite as

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

  • Maria Manuel Clementino
  • Dirk Hofmann
  • Walter Tholen
Article

Abstract

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad 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.

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 

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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Maria Manuel Clementino
    • 1
  • Dirk Hofmann
    • 2
  • Walter Tholen
    • 3
  1. 1.Departamento de MatemáticaUniversidade de CoimbraCoimbraPortugal
  2. 2.Departamento de MatemáticaUniversidade de AveiroAveiroPortugal
  3. 3.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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