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Here's a fun idea I had: dendrochronology (tree ring dating) is like sheaf theory. The idea is that trees in the same area grow the same pattern in their rings, aligned by year. So you start from a bunch of trees whose ages you know, and you try to match up overlaps between their patterns, in order to build a complete timeline
Ooh they're like stalks of an environmental sheaf! I like this a lot.
I didn't realise when I wrote this that it contained a pun, now I like it even more.