Maps I: relative to a factorisation system

Abstract

Originally, the categorical calculus of relations was developed using the canonical factorisation in regular categories. More recently, relations restricted to a proper factorisation system have been studied by several authors. In the present paper, we consider the general situation, in which relations are induced by an arbitrary stable factorisation. This extension of the calculus of relations is necessary for a categorical development of strongly constructive (and computational) logic, where non-monic relations come about naturally. In this setting, we analyse the correspondence of the maps, i.e. the total, single-valued relations, and the functions, as given by the arrows in the underlying category.