Filters and G-convergence in Categories

TL;DR

This paper develops a categorical framework for convergence and neighborhood concepts inspired by classical filter theory, extending these ideas to finitely complete categories to analyze sieves and their limits.

Contribution

It introduces the notions of filters, G-neighborhoods, and cover-neighborhoods in finitely complete categories, generalizing classical convergence concepts.

Findings

Defines categorical filters and neighborhoods analogous to classical topology.

Establishes a framework for studying convergence, cluster points, and closures in categories.

Provides tools for analyzing sieves and their limits in categorical contexts.

Abstract

In analogy with the classical theory of filters, for finitely complete categories, we provide the concepts of filter, G-neighborhood (short for \Grothendieck-neighborhood") and cover-neighborhood of a point, with the aim of studying convergence, cluster point and closure of sieves on objects of that kind of categories.