Abstract
In a locally \(\lambda \)-presentable category, with \(\lambda \) a regular cardinal, classes of objects that are injective with respect to a family of morphisms whose domains and codomains are \(\lambda \)-presentable, are known to be characterized by their closure under products, \(\lambda \)-directed colimits and \(\lambda \)-pure subobjects. Replacing the strict commutativity of diagrams by “commutativity up to \(\mathcal {\varepsilon }\)”, this paper provides an “approximate version” of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed \(\mathcal {\varepsilon }\)-generalizations of the notion of \(\lambda \)-purity. The categorical theory is being applied to the locally \(\aleph _1\)-presentable category of Banach spaces and their linear operators of norm at most 1, culminating in a largely categorical proof for the existence of the so-called Gurarii Banach space.
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Communicated by M. M. Clementino.
J. Rosický: Supported by the Grant Agency of the Czech Republic under the Grant P201/12/G028. W. Tholen: Supported by the Natural Sciences and Engineering Research Council of Canada under the Discovery Grants program.
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Rosický, J., Tholen, W. Approximate Injectivity. Appl Categor Struct 26, 699–716 (2018). https://doi.org/10.1007/s10485-017-9510-2
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DOI: https://doi.org/10.1007/s10485-017-9510-2
Keywords
- Met-enriched category
- Locally \(\lambda \)-presentable
- \(\mathcal {\varepsilon }\)-(co)limit
- \(\lambda \)-\(\mathcal {\varepsilon }\)-pure morphism
- \(\mathcal {\varepsilon }\)-injective object
- Approximate \(\lambda \)-injectivity class
- Urysohn space
- Gurarii space