A note on the exact completion of a regular category, and its infinitary generalizations

A new description of the exact completion $\cal C_{ex/reg}$ of a regular category $\cal C$ is given, using a certain topos $Shv(\cal C)$ of sheaves on $\cal C$; the exact completion is then constructed as the closure of $\cal C$ in $Shv(\cal C)$ under finite limits and coequalizers of equivalence relations. An infinitary generalization is proved, and the classical description of the exact completion is derived.