Approximate injectivity and smallness in metric-enriched categories
| Authors | |
|---|---|
| Year of publication | 2022 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Pure and Applied Algebra |
| MU Faculty or unit | |
| Citation | |
| Web | https://www.sciencedirect.com/science/article/pii/S0022404921003157?via%3Dihub |
| Doi | http://dx.doi.org/10.1016/j.jpaa.2021.106974 |
| Keywords | Metric enriched category; Approximate injectivity; Category of Banach spaces; Gurarii space |
| Description | 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 ?-(co)equalizers are given by ?-(co)isometries for all ?. 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. |
| Related projects: |