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Approximate injectivity

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ROSICKÝ Jiří THOLEN Walter

Rok publikování 2018
Druh Článek v odborném periodiku
Časopis / Zdroj Applied Categorical Structures
Fakulta / Pracoviště MU

Přírodovědecká fakulta

Citace
www http://dx.doi.org/10.1007/s10485-017-9510-2
Doi http://dx.doi.org/10.1007/s10485-017-9510-2
Klíčová slova enriched category; locally presentable category; pure morphism; injective object; approximate injectivity class; Urysohn space; Gurarii space
Popis 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 $\eps$", this paper provides an ``approximate version" of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed $\eps$-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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