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}




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



Has this "backwards" perspective on toposes been studied?
Actually, the closure operator of a topology is a finite *colimit* preserving monad on a powerset.


