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 this restriction, 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 and 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 huge 2-category of all regular categories.
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