Apologies for the vagueness of question.

Background

this thread has some nice examples of presheaves failing to be sheaves.

Question

Is there a generic way to measure "how badly" a presheaf fails at being a sheaf?

Something like an invariant that "counts", up to some notion of equivalence, sections that fail to glue or restrict properly?

Discussion

Can we do this by comparing some invariant of a presheaf $P$ and its sheaffification $\tilde P$? The comments on this old math SE thread makes an attempt to argue that the Cech cohomology (taking the cover refinement limit) of the two are equal. But is there something else that we can compare between $P$ and $\tilde P$?