Double category of relations in a locale [mde-U4GS]

Let \(L\) be a meet-semilattice. The double category \(\mathbb {R}\mathbf {el}(L)\) has data

• Objects: the elements of L,
• Loose morphisms \(U \to V\) are elements \(W \leqslant U \wedge V\),
• Tight morphisms given the order in L,
• Cells: exist when \(W \leqslant W'\) for the loose boundaries.

Composition of loose morphisms is given by \(\wedge \).

This is a special case of the double category of relations in a regular category.