In analogy with the relation between closure operators in presheaf toposes and Grothendieck topologies, we identify the structure in a category with finite limits that corresponds to universal closure operators in its regular and exact completions. The study of separated objects in exact completions will then allow us to give conceptual proofs of local cartesian closure of different categories of pseudo equivalence relations. Finally, we characterize when certain categories of sheaves are toposes.
Keywords: Exact completions, closure operators, toposes.
2000 MSC: 18A35, 18F10, 18B25.
Theory and Applications of Categories, Vol. 8, 2001, No. 21, pp 522-540.