Hi, I'm currently reading through "Sheaves in Geometry and logic" by Mclane-Moerdijk and this one issue has been bugging me for a long time, which I hope you could help me resolve.
It is known that for a general presheaf on a Grothendieck Topology, we must in general apply the plus construction twice to obtain a sheaf. The first application turns an arbitrary presheaf into a separated presheaf, and one more application gives a sheaf. So, exactly, intuitively, what obstructs us from getting a sheaf from just one application of the plus construction when the sheaf is non-separated? Let us take the example: What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf? given by Sherry. When we apply it once we get a separated presheaf, OK. But what exact component of that presheaf hindered us from getting the sheaf we wanted? I agree with what Sherry wrote in that case, namely that : "So in our example, 1 and 3, over ABC and BCD, in our original presheaf were compatible on a refinement of BC but not on BC" Can we generalize this notion to become rigorous in the case for arbitrary non-separated presheaves?