Grothendieck-topological Group Objects

TL;DR

This paper extends the classical theory of topological groups to finitely complete categories with Grothendieck topologies, introducing new concepts like G-topological spaces and group objects.

Contribution

It develops a categorical framework for G-topological groups, generalizing classical topological group theory to enriched categories with Grothendieck topologies.

Findings

Defined localized G-topological spaces and continuous morphisms.

Established the concepts of G-topological monoids and groups.

Provided a categorical foundation for G-topological group objects.

Abstract

In analogy with the classical theory of topological groups, for finitely complete categories enriched with Grothendieck topologies, we provide the concepts of localized G-topological space, initial Grothendieck topologies and continuous morphisms, in order to obtain the concepts of G-topological monoid and G-topological group objects.