The glueing construction and double categories

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.