The Glueing Construction and Double Categories

TL;DR

This paper develops a unifying framework using Artin-Wraith glueing to analyze exponentiability of locally closed inclusions across various mathematical structures like locales, toposes, and topological spaces.

Contribution

It introduces Artin-Wraith glueing in double categories and proves a general theorem on exponentiability of locally closed inclusions in multiple contexts.

Findings

Locally closed inclusions are exponentiable under certain conditions.

Exponentials can be constructed via Artin-Wraith glueing.

A unified theorem applies to five different mathematical settings.

Abstract

We introduce Artin-Wraith glueing and locally closed inclusions in double categories. Examples include locales, toposes, topological spaces, categories, and posets. With appropriate assumptions, we show that locally closed inclusions are exponentiable, and the exponentials are constructed via Artin-Wraith glueing. Thus, we obtain a single theorem establishing the exponentiability of locally closed inclusions in these five cases.