On closure operators, reflections and protolocalisations in Goursat categories

By defining a closure operator on effective equivalence relations in a regular category C, it is possible to establish a bijective correspondence between these closure operators and the regular epireflective subcategories of C, on the model of the closure operators on kernels in homological categories [5]. When C is an exact Goursat category [6], this correspondence restricts to a bijection between the Birkhoff closure operators on effective equivalence relations and the Birkhoff subcategories of C [2]. In this case it is possible to provide an explicit description of the closure, and this formula is used to describe the closure determined by the reflection of the category T(HComp) of compact Hausdorff Mal’cev algebras into its subcategory T(Profin) of profinite Mal’cev algebras. By using a result of Bourn [4], it is also possible to characterise the congruence distributive Goursat categories in terms of a property of the closure operator associated with any Birkhoff subcategory. In the second part of the talk we shall restrict our attention to the so-called “protolocalisations” [1]. In particular, we shall present a new characterisation of epireflective protolocalisations of an exact Mal’cev category, and give some examples [3].