TITLE: Sheafification - Why does twice suffice?
QUESTION [15 upvotes]: 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?

REPLY [19 votes]: "Why" questions can often be answered in multiple ways; I'll give an answer that's different from the other good comments and answers that you've already had.  
Sheaves are the first rung on an infinite ladder of concepts.  The next rung is "stack".  A stack is something like a "sheaf of categories", but there are added complications.  A typical example of a stack on a topological space $X$ is the assignment $U \mapsto Sh(U)$, sending each open subset $U \subseteq X$ to the category $Sh(U)$ of sheaves on $U$.  Sheaves on subsets of $X$ can be patched together, not quite uniquely, but uniquely up to canonical isomorphism.  From this, you can work out what the definition of stack must be.  This appears as an exercise somewhere in Mac Lane and Moerdijk.  
After stacks, there are 2-stacks (something like sheaves of 2-categories), 3-stacks, and so on.  (According to this terminological scheme, stacks are 1-stacks and sheaves are 0-stacks.)  And just as sheaves are presheaves satisfying certain conditions, $n$-stacks are "$n$-prestacks" satisfying certain conditions.
The point now is that "twice suffices" fits into a larger pattern.  For stacks, you need to apply (a version of) the plus construction three times.  For 2-stacks, you need to apply it four times.  In general, for $n$-stacks, you need to apply it $n+2$ times.
(I confess that this is something I've only heard in conversation, I believe from Andrew Kresch. I don't know the details.)