I’d hoped to have a pretty substantial post today. I fell short of having time to edit the beast into shape. I apologize but hope to have that soon.
I also hope to soon have an announcement about a Mathematics A-to-Z for this year. But until then, here’s this.
Several years ago in an A-to-Z I tried to explain cohomologies. I wasn’t satisfied with it, as, in part, I couldn’t think of a good example. You know, something you could imagine demonstrating with specific physical objects. I can reel off definitions, once I look up the definitions, but there’s only so many people who can understand something from that.
Quanta Magazine recently ran an article about homologies. It’s a great piece, if we get past the introduction of topology with that doughnut-and-coffee-cup joke. (Not that it’s wrong, just that it’s tired.) It’s got pictures, too, which is great.
This I came to notice because Refurio Anachro on Mathstodon wrote a bit about it. This in a thread of toots talking about homologies and cohomologies. The thread at this link is more for mathematicians than the lay audience, unlike the Quanta Magazine article. If you’re comfortable reading about simplexes and linear operators and multifunctions you’re good. Otherwise … well, I imagine you trust that cohomologies can take care of themselves. But I feel better-informed for reading the thread. And it includes a link to a downloadable textbook in algebraic topology, useful for people who want to give that a try on their own.