Defines Cauchy convergence and cocompleteness in V-normed categories via enrichment over normed sets and proves existence of Cauchy cocompletions plus a Banach fixed point theorem under light extra properties on V.
Kubi\'s , Categories with norms , preprint, arxiv.org/abs/1705.10189 https://arxiv.org/abs/1705.10189
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abstract
We study a metric-like structure on categories, showing that the concept of the limit of a sequence in a metric space and the concept of the colimit of a sequence in a category have a common generalization. The main concept is a norm on a category, generalizing pseudo-metrics and group valuations. In this new context, we discuss topics like Cauchy completion and the Banach Contraction Principle.
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UNVERDICTED 2representative citing papers
The set of universal and ultrahomogeneous retractions on the Urysohn space is characterized topologically via a new extension property (UR*) equivalent to those features and a new pointwise retract topology, yielding results on its Borel complexity and density.
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Cauchy convergence in V-normed categories
Defines Cauchy convergence and cocompleteness in V-normed categories via enrichment over normed sets and proves existence of Cauchy cocompletions plus a Banach fixed point theorem under light extra properties on V.
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Topological size of the set of universal and ultrahomogeneous retractions on the Urysohn space
The set of universal and ultrahomogeneous retractions on the Urysohn space is characterized topologically via a new extension property (UR*) equivalent to those features and a new pointwise retract topology, yielding results on its Borel complexity and density.