Abstract
Due to the nature of compactness, there are several interesting ways of defining compact objects in a category. In this paper we introduce and study an internal notion of compact objects relative to a closure operator (following the Borel-Lebesgue definition of compact spaces) and a notion of compact objects with respect to a class of morphisms (following Áhn and Wiegandt [2]). Although these concepts seem very different in essence, we show that, in convenient settings, compactness with respect to a class of morphisms can be viewed as Borel-Lebesgue compactness for a suitable closure operator. Finally, we use the results obtained to study compact objects relative to a class of morphisms in some special settings.
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Partial financial assistance by Centro de Matemática da Universidade de Coimbra and by a NATO Collaborative Grant (CRG 940847) is gratefully acknowledged.
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Clementino, M.M. On categorical notions of compact objects. Appl Categor Struct 4, 15–29 (1996). https://doi.org/10.1007/BF00124111
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DOI: https://doi.org/10.1007/BF00124111