Filters and compactness on small categories and locales

Abstract

In analogy with the classical theory of filters, for fi\-nite\-ly complete or small cat\-e\-go\-ries, we provide the concepts of fil\-ter, $\mathfrak{G}$-neigh\-bor\-hood (short for "Grothendieck-neigh\-bor\-hood") and cover-neigh\-bor\-hood of points of such categories, with the goal of studying convergence, cluster point, closure of sieves and compactness on objects of that kind of categories. Finally, we study all these concepts in the category $\bf{Loc}$ of locales.