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 1For 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 2For 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?

colimitpreserving monad on a powerset? Two disjoint sets can have closure points in common, so closure doesn't preserve intersections. $\endgroup$1more comment