Lawvere Completeness in Topology |
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Authors: | Maria Manuel Clementino Dirk Hofmann |
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Institution: | (1) CMUC/Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal;(2) UIMA/Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal |
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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
(\mathbbT,V)(\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 V\mathsf{V} has a canonical
(\mathbbT,V)(\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
(\mathbbT,V)(\mathbb{T},\mathsf{V})-categories. |
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Keywords: | -category" target="_blank">gif" alt="$\mathsf{V}$" align="middle" border="0">-category Bimodule Monad gif" alt="$(\mathbb{T} -category" target="_blank">\mathsf{V})$" align="middle" border="0">-category Completeness |
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