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@article{1805238,
author = {Adámek, Jiří and Rosický, Jiří},
article_number = {6},
doi = {http://dx.doi.org/10.1016/j.jpaa.2021.106974},
keywords = {Metric enriched category; Approximate injectivity; Category of Banach spaces; Gurarii space},
language = {eng},
issn = {0022-4049},
journal = {Journal of Pure and Applied Algebra},
title = {Approximate injectivity and smallness in metric-enriched categories},
url = {https://www.sciencedirect.com/science/article/pii/S0022404921003157?via%3Dihub},
volume = {226},
year = {2022}
}
TY - JOUR
ID - 1805238
AU - Adámek, Jiří - Rosický, Jiří
PY - 2022
TI - Approximate injectivity and smallness in metric-enriched categories
JF - Journal of Pure and Applied Algebra
VL - 226
IS - 6
SP - 106974
EP - 106974
PB - Elsevier
SN - 00224049
KW - Metric enriched category
KW - Approximate injectivity
KW - Category of Banach spaces
KW - Gurarii space
UR - https://www.sciencedirect.com/science/article/pii/S0022404921003157?via%3Dihub
N2 - 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.
ER -
ADÁMEK, Jiří a Jiří ROSICKÝ. Approximate injectivity and smallness in metric-enriched categories. \textit{Journal of Pure and Applied Algebra}. Elsevier, 2022, roč.~226, č.~6, s.~106974-107003. ISSN~0022-4049. Dostupné z: https://dx.doi.org/10.1016/j.jpaa.2021.106974.
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