Double Categories of Relations

Abstract

A double category of relations is essentially a cartesian equipment with strong, discrete and functorial tabulators and for which certain local products satisfy a Frobenius Law. A double category of relations is equivalent to a double category whose proarrows are relations on some ordinary category admitting a proper and stable factorization system. This characterization is based closely on the recent characterization of double categories of spans due to Aleiferi. The overall development can be viewed as a double-categorical version of that of the notion of a "tabular allegory" or that of a "functionally complete bicategory of relations."