Interor and h operators on the category of locales

TL;DR

This paper introduces interior and h operators on the category of locales, constructing associated topological categories and exploring their properties to advance the categorical understanding of locales.

Contribution

It defines interior and h operators on locales and constructs new topological categories based on these operators, expanding the categorical framework for locales.

Findings

Construction of the topological category ( extbf{I-}Loc) with interior operators.

Introduction of extit{h} operators and their properties.

Development of associated topological categories for locales.

Abstract

We present the concept of interior operator $I$ on the category $\mathbf{Loc}$ of locales and then we construct a topological category \linebreak $\big(\mathbf{I\text{-}Loc},\ U\big)$, where $U:\mathbf{I\text{-}Loc}\rightarrow \mathbf{Loc}$ is a forgetful functor; and we also introduce the notion of $\mathfrak h$ operator on the category $\mathbf{Loc}$ and discuss some of their properties for constructing the topological category $\big(\mathbf{\mathfrak h\text{-}Loc},\ U\big)$ associated to the forgetful functor $U:\mathbf{\mathfrak h\text{-}Loc}\rightarrow \mathbf{Loc}$.