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Applied Categorical Structures

, Volume 17, Issue 2, pp 175–210 | Cite as

Lawvere Completeness in Topology

  • Maria Manuel Clementino
  • Dirk Hofmann
Article

Abstract

It is known since 1973 that Lawvere’s notion of Cauchy-complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper, we introduce the corresponding notion of Lawvere completeness for \((\mathbb{T},\mathsf{V})\)-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones it means weak sobriety while for the latter it means Cauchy completeness. Further, we show that \(\mathsf{V}\) has a canonical \((\mathbb{T},\mathsf{V})\)-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; this structure permits us to define a Yoneda embedding in the realm of \((\mathbb{T},\mathsf{V})\)-categories.

Keywords

\(\mathsf{V}\)-category Bimodule Monad \((\mathbb{T},\mathsf{V})\)-category Completeness 

Mathematics Subject Classifications (2000)

18A05 18D15 18D20 18B35 18C15 54E15 54E50 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.CMUC/Department of MathematicsUniversity of CoimbraCoimbraPortugal
  2. 2.UIMA/Department of MathematicsUniversity of AveiroAveiroPortugal

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