The double category of relations is one of the most beloved examples of double categories, with many variations, such as frame-valued relations and relations with respect to different kinds of monomorphisms (the right class of a factorization system). Such double categories are systematically captured by Shulman’s Fr construction, which produces a double category of relations internal to a certain class of monoidal fibrations.
In this talk, I will present a generalization of this construction that produces a virtual double category of relations, where loose morphisms are not required to compose. I will also give a characterization of the class of fibrations called elementary existential fibrations as those for which the induced virtual double category is a cartesian equipment. In fact, these fibrations and cartesian equipments with one additional assumption form biequivalent 2-categories. If time permits, I will discuss a further generalization of this construction that captures more than binary relations, which is ongoing joint work with Aaron Fairbanks.