Relations in Categories

This thesis investigates relations over a category C relative to an (E;M)-factorization system of C. In order to establish the 2-categoryRel(C) of relations over C in the rst part we discuss su cient conditions for the associativity of horizontal composition of relations, and we investigate special classes of morphisms in Rel(C). Attention is particularly devoted to the notion of mapping as de ned by Lawvere. We give a signi cantly simpli ed proof for the main result of Pavlovi c, namely that C ' Map(Rel(C)) if and only if E RegEpi(C). This part also contains a proof that the category Map(Rel(C)) is nitely complete, and we present the results obtained by Kelly, some of them generalized, i. e., without the restrictive assumption that M Mono(C). The next part deals with factorization systems in Rel(C). The fact that each set-relation has a canonical image factorization is generalized and shown to yield an ( E; M)-factorization system in Rel(C) in case M Mono(C). The setting without this condition is studied, as well. We propose a weaker notion of factorization system for a 2-category, where the commutativity in the universal property of an (E;M)-factorization system is replaced by coherent 2-cells. In the last part certain limits and colimits in Rel(C) are investigated. Coproducts exist inRel(C) and are given as in C provided that C is extensive. However, nite (co)completeness fails. Finally we show that colimits of !-chains do not exist in Rel(C) in general. However, it turns out that a canonical construction with a 2-categorial universal property exists if C has well-behaved colimits of !-chains. For the case E Epi(C) we give a necessary and su cient condition that forces our construction to yield colimits of !-chains in Map(Rel(C)).