Filters and compactness on small categories and locales

TL;DR

This paper develops a theory of filters, neighborhoods, and compactness for small categories and locales, extending classical ideas to categorical and locale settings.

Contribution

It introduces new concepts of filters and neighborhoods in small categories and locales, providing a framework for convergence and compactness in these structures.

Findings

Defined filters, neighborhood concepts, and cover-neighborhoods in small categories.

Analyzed convergence, cluster points, and closure of sieves within these categories.

Studied the properties of these concepts specifically in the category of 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.