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- W3215071638 abstract "Properties of categories enriched over the category of metric spaces are investigated and applied to a study of well-known constructions of metric and Banach spaces. We prove e.g. that weighted limits and colimits exist in a metric-enriched category iff ordinary limits and colimits exist and ε-(co)equalizers are given by ε-(co)isometries for all ε. An object is called approximately injective w.r.t. a morphism h:A→A′ iff morphisms from A into it are arbitrarily close to those morphisms that factorize through h. We investigate classes of objects specified by their approximate injectivity w.r.t. given morphisms. They are called approximate-injectivity classes. And we also study, conversely, classes of morphisms specified by the property that certain objects are approximately injective w.r.t. them. For every class of morphisms satisfying a mild smallness condition we prove that the corresponding approximate-injectivity class is weakly reflective, and we study the properties of the reflection morphisms. As an application we present a new categorical proof of the essential uniqueness of the Gurarii space." @default.
- W3215071638 created "2021-12-06" @default.
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- W3215071638 date "2022-06-01" @default.
- W3215071638 modified "2023-10-17" @default.
- W3215071638 title "Approximate injectivity and smallness in metric-enriched categories" @default.
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- W3215071638 doi "https://doi.org/10.1016/j.jpaa.2021.106974" @default.
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