Abstract
A number of authors have observed that regularity of a category A is not necessary for the existence of a "calculus of relations" in A, with an associative composition of relations giving a 2-category Rel A; it suffices that the finitely-complete A have a proper factorization system (ε,M) whose class ε is stable by pullbacks, the classical regular-category case being that where M consists of all the monomorphisms. We show that this generalization is in a sense illusory: if B is the category of "maps" in Rel A, then B is a regular category, and Rel A is isomorphic to the classical Rel B.
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