On categorical notions of compact object
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 Ahn and Wiegandt).
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.
