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Cartesian closed topological hulls as injective hulls. (English) Zbl 0614.18003
Topological hulls are characterized as injective hulls in the quasicategory of the fibre-small concrete categories over an arbitrary category and it has been desired to give similar results for Cartesian closed topological hulls. In this paper the authors accomplish this by introducing the concept of concretely Cartesian closed topological hulls, as follows.
Let X be a Cartesian closed category. In the quasicategory \(Cat_ PX\) whose objects are the small fibred concrete categories over X with finite concrete products and whose morphisms are the concrete functors over X that preserve these products, injective hulls are concretely Cartesian closed topological hulls. In case X is the terminal category, this means that injective hulls are local hulls in the category of (meet-) semilattices, the result by Bruns-Lakser and Horn-Kimura. In case X is Set, the authors also give the characterization of Cartesian closed topological hulls in the quasicategory of fibre-small concrete categories over Set with certain additional but natural conditions.
Reviewer: R.Nakagawa

MSC:
18A99 General theory of categories and functors
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
18G05 Projectives and injectives (category-theoretic aspects)
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
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