some brave people, they came back the next day, and tried to look into the house in a different way. it was hard to see , but the light was on, and there was a visitor. the visitor was listening to an explanation of why sheaf is a set of things which vary continuously.

R valuation ring

f : X  →

if f is flat, sheaf of of regular functions on X. then the fibers $ f^{-1}(s)$ vary continuously.
(overheard in a different room) this is actually simpler to understand than what is written in Mac Lane and Moerdijk. 
all ideals $\mathfrak{a} \in R$. the visitor said, 'why valuation ring R?', and the reply was '... generic closed points of Spec R'. (overheard while visitor was leaving). '... module of syzygies'. $f : X \rightarrow Y$