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Recently, as a final degree project, I have been studying basic category theory. My tutor is very experienced in monads and so I am studying them. I have been researching some curious examples (the Giry monad, the monad of Directed graphs, the powerset monad, monads that come from a free-forgetful scenario where you forget the additional structure, the monad that "kills" the torsion)...

But I am very curious to know where could I find monads in topology? Or in analysis? What are the strangest monads you know?

I have consulted this answer but I knew most of them. I am looking for something less abstract. I know my professor told me that you could see the real numbers as a monad. Something like that.

Aliara
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  • Depending on your background, my answers to this question and also this MO question could be of interest to you. – Arnaud D. May 23 '18 at 10:27
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    In topology, the ultrafilter monad (over $Set$) has very interesting algebras : they're precisely the compact Hausdorff spaces – Maxime Ramzi May 23 '18 at 13:20
  • "monads that come from a free-forgetful scenario" - if you generalize this to pairs of adjoint functors, then every monad can be expressed as a special case of this (either using the Eilenberg-Moore category, or the Kleisli category). – Daniel Schepler Oct 03 '18 at 23:42
  • In algebraic geometry, the Godement functor that takes a sheaf to a certain flasque "extension" of that sheaf is a monad. – Daniel Schepler Oct 03 '18 at 23:45
  • I am with @Max. I am currently working on the ultrafilter monad, using the ultrafilter space defined by Salbany. It's quite interesting. You can even get construct its algebra using the retraction of the ultrafilter space. – Percy Oct 03 '18 at 23:07
  • @DanielSchepler Every monad comes from a free-forgetful scenario, if you consider the Eilenber-Moore category as "extra structure". – Arnaud D. Oct 05 '18 at 09:57

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