When do completion processes give rise to extensive categories?

Lack, S;Vitale, Enrico
(2001) Journal of Pure and Applied Algebra — Vol. 159, n° 2-3, p. 203-230 (2001)

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We consider various (free) completion processes: the exact completion and the regular completion of a category with weak finite limits, the pre-regular completion of a category with finite products and weak finite limits, the exact completion of a regular category, the regular reflection of a pre-regular category, and the filtered-colimit completion of a small category. In each case we give necessary and sufficient conditions for the completion to be extensive; or, in the case of the pre-regular completion, for the completion to satisfy a weakened notion of extensivity which we call pre-extensivity. (C) 2001 Elsevier Science B.V. All rights reserved.
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Lack, S., & Vitale, E. (2001). When do completion processes give rise to extensive categories? Journal of Pure and Applied Algebra, 159(2-3), 203-230. https://doi.org/10.1016/S0022-4049(00)00060-8 (Original work published 2001)