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

Abstract

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.