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

We consider the ordinary category Span (C) of (isomorphism classes of) spans of morphisms in a category C with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of Span (C) to be an allegory. In particular, when C carries a pullback-stable, but not necessarily proper, (E , M) -factorization system, we establish a quotient category Span E (C) that is isomorphic to the category Rel M (C) of M -relations in C , and show that it is a (unitary and tabular) allegory precisely when M is a class of monomorphisms in C . Without the restriction to monomorphisms, one can still find a least pullback-stable and composition-closed class E ∙ containing E such that Span E ∙ (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. [ABSTRACT FROM AUTHOR]