Apologies for the vagueness of question.

**Background**

[this thread](https://mathoverflow.net/a/45218/74739) 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 sheafification $\tilde P$? The comments on [this old math SE thread](https://math.stackexchange.com/questions/803551/%C4%8Cech-cohomology-with-values-in-a-presheaf) 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 $\tilde P$?