# Has this “backwards” perspective on toposes been studied?

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?

• This is really self promotion, but your question is related to my talk at CT: math.muni.cz/~diliberti/Talk/Scott.pdf – Ivan Di Liberti Sep 25 '19 at 21:24
• 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. – Mike Shulman Oct 20 '19 at 14:55
• @MikeShulman I think that's the answer! How embarrassing. – Oscar Cunningham Oct 20 '19 at 16:11
• @MikeShulman Add that as an answer? – Oscar Cunningham Dec 2 '19 at 12:02
• @IvanDiLiberti Your link is broken :( – Martin Brandenburg Jan 10 at 21:58