When do completion processes give rise to extensive categories? (Q5939822)

In addition to providing an excellent summary on extensive categories and on various completion processes for categories, with a useful and up-to-date list of references, the authors give a comprehensive answer to the question given by the title of their paper: which conditions on a category \({\mathcal C}\) with finite coproducts are (necessary and) sufficient in order for a ``completion'' \({\mathcal C}^\wedge\) of \({\mathcal C}\) to be extensive (so that in the formulation of Schanuel, \({\mathcal C}^\wedge\) has finite coproducts and the obvious functor \({\mathcal C}^\wedge/X \times {\mathcal C}^\wedge/Y\to{\mathcal C}^\wedge/ (X+Y)\) is an equivalence of categories, for all objects \(X,Y)\)?NEWLINENEWLINENEWLINEHere ``completion'' may mean, for example, theNEWLINENEWLINENEWLINE-- Cauchy completion of \({\mathcal C}\) (which makes idempotents split)NEWLINENEWLINENEWLINE-- filtered-colimit completion when \({\mathcal C}\) is Cauchy completeNEWLINENEWLINENEWLINE-- (Barr-)exact completion when \({\mathcal C}\) is regularNEWLINENEWLINENEWLINE-- regular completion when \({\mathcal C}\) has weak finite limits NEWLINENEWLINENEWLINE-- regular completion when \({\mathcal C}\) is preregular (= finitely complete with a proper stable factorization system)NEWLINENEWLINENEWLINE-- preregular completion when \({\mathcal C}\) has finite products and weak equalizers.