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Semilocalizations of exact and leftextensive categories. (English) Zbl 0982.18003

A full replete subcategory of a category with finite limits is called a semilocalization if the reflector functor preserves pullbacks along arrows that lie in the subcategory. This paper shows that semilocalizations of regular categories are regular and that semilocalizations of locally distributive categories are locally distributive. In addition, it is shown that a semilocalization of an exact category has coequalizers of equivalence relations that are stable under pullbacks; moreover, this is diagnostic in the sense that a regular category is a semilocalization of its exact completion if and only if it has stable coequalizers of equivalence relations. Similarly, any locally distributive category is a semilocalization of its free sum completion. The paper also studies simliar results for some classes of elementary-topos-like categories.

MSC:

18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18E35 Localization of categories, calculus of fractions
18E10 Abelian categories, Grothendieck categories
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18A22 Special properties of functors (faithful, full, etc.)
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