Filters and compactness on small categories and locales

In analogy with the classical theory of filters, for finitely complete or small categories, we provide the concepts of filter, G -neighborhood (short for “Grothendieck-neighborhood”) and cover-neighborhood of points of such categories, to study convergence, cluster point, closure of sieves and compactness on objects of that kind of categories. Finally, we study all these concepts in the category L o c of locales.