TITLE: Has this "backwards" perspective on toposes been studied?
QUESTION [15 upvotes]: 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?

REPLY [3 votes]: Actually, the closure operator of a topology is a finite colimit preserving monad on a powerset.