Quotients of Span Categories that are Allegories and the Representation of Regular Categories

We consider the ordinary category \(\mathsf {Span}({\mathcal {C}})\) of (isomorphism classes of) spans of morphisms in a category \(\mathcal {C}\) with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of \(\mathsf {Span}({\mathcal {C}})\) to be an allegory. In particular, when \({\mathcal {C}}\) carries a pullback-stable, but not necessarily proper, \(({\mathcal {E}},{\mathcal {M}})\)-factorization system, we establish a quotient category \(\mathsf {Span}_{{\mathcal {E}}}({\mathcal {C}})\) that is isomorphic to the category \(\mathsf {Rel}_{{\mathcal {M}}}({\mathcal {C}})\) of \({\mathcal {M}}\)-relations in \({\mathcal {C}}\), and show that it is a (unitary and tabular) allegory precisely when \({\mathcal {M}}\) is a class of monomorphisms in \({\mathcal {C}}\). Without the restriction to monomorphisms, one can still find a least pullback-stable and composition-closed class \({\mathcal {E}}_{\bullet }\) containing \(\mathcal E\) such that \(\mathsf {Span}_{{\mathcal {E}}_{\bullet }}({\mathcal {C}})\) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the 2-category of all regular categories.