I’ve just started writing my revision notes for Riemann surfaces, and it’s suddenly dawned on me that it gives a hazy explanation of why the sheaf cohomology might give the same answer as the kind of homology theory you get by messing about with paths.

If you take a function defined on a little ball in your space, and then analytically continue it along a path, you get a unique analytic continuation. However, if you take a path, a loop say, which corresponds to nontrivial homology (like going around zero when you’ve got hold of a branch of the logarithm), you end up collecting something you didn’t have when you started. In particular, you’re not extending to a well-defined global section. I like to believe that maybe we also have some kind of converse… namely, if you have a local section failing to be global, then a measure of that failure is the same as the number of different ‘loops’ you can analytically continue the function along (so maybe I want non-homologous paths to always give rise to different local sections in some sense).

Maybe once I’ve finished revising the course properly (and probably after exams when there might be time to think about things like sheaves and cohomology), I’ll have some more intelligent things to say (after all, now I think about it, that the etale space provides a universal cover, at least in the sense of analysis, is pretty suggestive… although this would suggest analogues closer to that of the fundamental group…. so something has to get Abelian somewhere…) but I thought this was worth recording. Perhaps the wonderful world of Riemann surfaces is the best way of leaving paths behind and accepting local function rings as the gatekeepers to the geometry of a space… if indeed they are.