Abstract
The regular and exact completions of categories with weak limits are proved to exist and to be determined by an appropriate universal property. Several examples are discussed, and in particular the class of examples given by categories monadic over a power of Set: any such a category is in fact the exact completion of the full subcategory of free algebras. Applications to Grothendieck toposes and geometric morphisms, and to epireflective hulls are also discussed.
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