Blessed theorem https://twitter.com/dzackgarza/status/1264023160023912449
I know this definition because it's the one that translates into algebraic geometry, but the diagonal shows up in all kinds of other places if you knew where to look. The idea is that it can tell when two maps into your space coincide.
For example, if X and Y are measurable spaces, and the diagonal of Y is a measurable set, then any two measurable maps X -> Y disagree on a measurable set. In particular if Y is Hausdorff and its algebra contains the Borel algebra, the result applies by the topological theorem
That's quite far where from where I would expect something arrow theoretic like this to pop up.

Something particularly nice happens you have a map f: X -> X. You can take the map X -> X x X given by w |-> (w, f(w)). Where does this intersect the diagonal? A fixed point!
This shows up a lot in differential topology... I just realized I am going to do a massive thread if I keep talking about the next interesting thing that occurs to me and I really need to do things so I'll briefly close.

Trace maps are also closely to checking for fixed points.
And the Euler characteristic is ALSO closely related to fixed points (it's controlled by the fixed points you are forced to get when deforming the identity)!
If you are very clever and know more diff top than me you could try using all these tools at once and get amazing results like the lefschetz fixed point theorem!

Ok no more rn
(Maybe at some point a Sarah will do a thread about recursion and the diagonal)
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