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I worked a little on the entry separated presheaf. Apart from some general editing I
added construction and proof of the separafication functor;
began a section on the full notion of bi-separated presheaf.
More deserves to be done here, but I have to stop for the moment.
It would be good to move biseparated to a separate entry. It is a much more rarely used concept.
Added the remark that a constant sheaf on a site for which every covering family is inhabited is in fact separated (as per my recent MO question/answer).
Thanks! I have cross-lined a bit with locally connected site.
If you find a minute, please consider either adding a pointer to the proof or the proof itself (simple as it may be, but for completeness).
Added:
Diffeological spaces are biseparated presheaves for the bisite structure on cartesian spaces, where $J$ is the usual topology of open covers and $K$ is the topology given by the families
$\coprod_{u\in U}\mathbf{R}^0 \to U$for every cartesian space $U$.
The sheaf condition with respect to $J$ yields a smooth set, whereas the separated presheaf condition with respect to $K$ makes it into a diffeological space.
Added a reference to epipresheaf.
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