This is crosspost of this MSE question.

Having carefully read the proof of Beck's monadicity theorems and some related variations, I'm now hungry for cool applications.

For instance, I found these blackboard pictures of a lecture by David Jordan, but I do not understand any of the examples - too terse for me.

What are some interesting applications of monadicity theorems, especially in geometric contexts (algebraic geometry would be especially cool)? Can anyone spell out the example(s) presented in the linked pictures?

Added. I know about monadic descent and am trying to slowly understand it. I am hoping for additional examples.

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    $\begingroup$ You're probably aware of this, but one of the early big surprises was the fact that compact Hausdorff spaces are monadic over sets. One consequence of this that I still find pretty amazing is that the category of compact Hausdorff spaces is a pretopos (being Barr exact and extensive). $\endgroup$ – Todd Trimble Mar 29 '17 at 19:14
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    $\begingroup$ @ToddTrimble I am aware but now understand I don't appreciate this fact enough. Thanks! $\endgroup$ – Arrow Mar 29 '17 at 23:07

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