# Factorizing the Top-Loc adjunction through positive topologies

## TL;DR

This paper characterizes positive topologies as a fibration over locales, constructs an adjunction with topological spaces, and shows how the classical Top-Loc adjunction factors through this new framework.

## Contribution

It introduces a novel categorical framework relating positive topologies, locales, and topological spaces via fibrations and adjunctions.

## Key findings

- Positive topologies form a fibration over locales.
- An adjunction between positive topologies and Top is constructed.
- The classical Top-Loc adjunction factors through the new adjunction.

## Abstract

We characterize the category of Sambin's positive topologies as a fibration over the category of locales Loc. The fibration is obtained by applying the Grothendieck construction to a doctrine over Loc. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the newly constructed adjunction.