Closure operators in exact completions (Q2756989)

It is well-known that for a small category \({\mathcal C}\) the universal closure operators on the presheaf topos \({\mathcal S}et^{\mathcal C}\) are in correspondence with the Grothendieck topologies on \({\mathcal C}\). This provides a framework for investigating separated objects and sheaves. The article under review explores these ideas in the context of exact completions. The reader will not only need familiarity with topologies, separated objects and sheaves, but also with regular and exact categories and completions.NEWLINENEWLINENEWLINEThe author proceeds to develop the notions of a topology on a category with finite limits, separated objects, canonical topologies, and sheaves. One of the main results of the paper establishes the equivalence between the following categories where \({\mathcal C}\) is a regular category and \({\mathcal C}_{\text{ex}}\) 2) denotes its exact completion: 1) sheaves for the canonical topology on \({\mathcal C}_{\text{ex}}\) the full subcategory of \({\mathcal C}_{ex}\), given by the complete equivalence relations, 3) the full subcategory of \({\mathcal C}_{\text{ex}}\) given by the Higgs-complete equivalence relations.NEWLINENEWLINENEWLINEThe last two sections of the paper discuss local Cartesian closure, and complete equivalence relations and toposes.