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Topos theory can be seen as a categorification of topology via the following analogies.

\begin{array}{|c|c|} \hline \text{locales}&\text{Grothendieck toposes}\\\hline \text{open sets}&\text{sheaves}\\\hline \text{continuous maps}&\text{geometric morphisms}\\\hline \text{bases}&\text{sites}\\\hline \text{topological spaces}&\text{ionads}\\\hline \end{array}

But when I was first learning about topos theory I was temporarily confused by the following two results.

Proposition 1

For a set $S$ there's a correspondence between topologies on $S$ and finite limit preserving (necessarily idempotent) monads on $\mathbf{2}^S$. A topology corresponds to its closure operation.

Proposition 2

For a category $C$ there's a correspondence between Grothendieck topologies on $C$ and finite limit preserving idempotent monads on $\mathbf{Set}^{C^\mathrm{op}}$. A Grothendieck topology corresponds to its sheafification operation.

The similarity between these propositions suggests that we can also view topos theory as a categorification of topology in such a way that sheaves are the categorification of closed sets rather than open ones. From this strange perspective the analogies would be the following.

\begin{array}{|c|c|} \hline \text{topologies}&\text{Grothendieck topologies}\\\hline \text{topological spaces}&\text{sites}\\\hline \text{closed sets}&\text{sheaves}\\\hline \text{continuous maps}&\text{morphisms of sites}\\\hline \end{array}


Has this point of view been studied anywhere? Does it have any use?

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    $\begingroup$ This is really self promotion, but your question is related to my talk at CT: math.muni.cz/~diliberti/Talk/Scott.pdf $\endgroup$ Sep 25, 2019 at 21:24
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    $\begingroup$ Isn't the closure operator of a topology a finite colimit preserving monad on a powerset? Two disjoint sets can have closure points in common, so closure doesn't preserve intersections. $\endgroup$ Oct 20, 2019 at 14:55
  • $\begingroup$ @MikeShulman I think that's the answer! How embarrassing. $\endgroup$ Oct 20, 2019 at 16:11
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    $\begingroup$ @MikeShulman Add that as an answer? $\endgroup$ Dec 2, 2019 at 12:02
  • $\begingroup$ @IvanDiLiberti Your link is broken :( $\endgroup$ Jan 10, 2020 at 21:58

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Actually, the closure operator of a topology is a finite colimit preserving monad on a powerset.

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  • $\begingroup$ Did you really have to start it that way? $\endgroup$ Dec 3, 2019 at 5:35
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    $\begingroup$ @DavidRoberts what? $\endgroup$ Dec 3, 2019 at 14:59
  • $\begingroup$ Reads to me like a "well, actually...". No biggie, but the answer doesn't need it. $\endgroup$ Dec 3, 2019 at 20:43
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    $\begingroup$ @DavidRoberts In my idiolect, the word "actually" is used when pointing out that the facts of a situation are different from what has previously been stated or assumed. I think it is appropriate and useful here because the "answer" is not really answering the question asked, but instead pointing out that it is based on an incorrect premise, and the reader's attention should be drawn to that fact. $\endgroup$ Dec 3, 2019 at 22:45

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