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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 $\endgroup$ – Ivan Di Liberti Sep 25 '19 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. $\endgroup$ – Mike Shulman Oct 20 '19 at 14:55
  • $\begingroup$ @MikeShulman I think that's the answer! How embarrassing. $\endgroup$ – Oscar Cunningham Oct 20 '19 at 16:11
  • $\begingroup$ @MikeShulman Add that as an answer? $\endgroup$ – Oscar Cunningham Dec 2 '19 at 12:02
  • $\begingroup$ @IvanDiLiberti Your link is broken :( $\endgroup$ – Martin Brandenburg Jan 10 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? $\endgroup$ – David Roberts Dec 3 '19 at 5:35
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    $\begingroup$ @DavidRoberts what? $\endgroup$ – Mike Shulman Dec 3 '19 at 14:59
  • $\begingroup$ Reads to me like a "well, actually...". No biggie, but the answer doesn't need it. $\endgroup$ – David Roberts Dec 3 '19 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. $\endgroup$ – Mike Shulman Dec 3 '19 at 22:45

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