(This is in a sense a follow-up to [this question](https://mathoverflow.net/questions/342466).)


I [was under the impression these days](https://mathoverflow.net/questions/437199) that Grothendieck topoi were also¹ analogous to topological spaces in that the former were left exact reflective localisations of presheaf categories $\mathsf{PSh}(\mathcal{C})=\mathsf{Cat}(\mathcal{C}^\mathsf{op},\mathsf{Set})$, while the latter were left exact reflective localisations of powersets $\mathcal{P}(X)=\mathsf{Set}(X,\{\mathrm{t},\mathrm{f}\})$.


However, I finally realised that the latter actually gives a slightly different notion:


* We have a subcategory $\mathsf{Open}'(X)$ of $\mathcal{P}(X)$;
* A left adjoint $L$ to the embedding $\iota\colon\mathsf{Open}'(X)\hookrightarrow\mathcal{P}(X)$, and hence:
	1. Preserving colimits, i.e. $L(\emptyset)=\emptyset$ and $L\big(\bigcup\_{i\in I}U\_i\big)=\bigcup\_{i\in I}L(U\_i)$;
	2. Satisying the universal property of the *closure* of a set, i.e.: for $S\in\mathcal{P}(X)$ and $T\in\mathsf{Open}'(X)$, we have $L(S)\subset T$ iff $S\subset T$.
* Finally, $L$ is left exact, and hence preserves finite limits, i.e. $L(X)=X$ and $L(U\_1\cup\cdots\cup U\_n)=L(U\_1)\cup\cdots\cup L(U\_n)$.


The sole two differences between the above and a topological space is that 1) $L$ satisfies the "wrong universal property" (that of the closure instead of the interior), and 2) $L$ in addition preserves arbitrary unions, which the interior does not.



> 
> **Question 1.** Does this decategorified notion of a Grothendieck topos lead to anything interesting? Has it already been studied before?
> 
> 
> Alternatively, we could also consider right exact coreflective localisations of $\mathcal{P}(X)$ (corresponding to "cosheaf 'co'topoi" in the 1-categorical case). What about these?
> 
> 
> 


Then, a second way to “decategory” Grothendieck topoi would be via monads/closure operators:


* A Grothendieck topology on $\mathcal{C}$ is the same thing as a finite limit preserving idempotent monad on $\mathsf{PSh}(\mathcal{C})$ (sheafification);
* A topology on $X$ is the same thing as a finite *colimit* preserving idempotent monad on $\mathcal{P}(X)$.



> 
> **Question 2.** Are finite **limit** preserving idempotent monads on $\mathcal{P}(X)$ of any interest? Again, are they actually already known structures under a different name?
> 
> 
> 


Finally, we could play the reverse game and try to category the known definitions of topologies:



> 
> **Question 3.** What about finite **colimit** preserving idempotent monads on $\mathsf{PSh}(\mathcal{C})$?
> 
> 
> 




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¹Grothendieck topoi are thought as being categorified locales, where the proper categorification of a topological space is an ionad ([nLab page](https://ncatlab.org/nlab/show/ionad), [original paper introducing them](https://arxiv.org/abs/0912.1415)); see [Kevin Arlin's answer here](https://mathoverflow.net/a/437204).



Decategorifying Grothendieck topoi and categorifying topological spaces
This type of structure is equivalently given by the choice of a subset $S\subset X$. One can give a topos-theoretic proof of that fact ( you are describing exactly a $(-1)$-topos, and left exact localizations of presheaf $n$-topoi are always topological when $n<\infty$), but also a very elementary one in this simpler case.


Let $L: P(X)\to P(X)$ be a left exact idempotent monad, and let $S \subset X$ be $L(0)$; and let $\delta\_p : x\mapsto [p=x]\in 2$ (this is just the Yoneda embedding for the discrete category $X$)


*Claim 1*: For $x\in S$, $L(\delta\_x) = L(0) = S$. Indeed, the map $0\to \delta\_x$ induces $L(0)\to L(\delta\_x)$. Furthermore, because $x\in S, \delta\_x\leq S= L(0)$, so that by adjunction, $L(\delta\_x)\leq L(0)$.


*Claim 2*: Conversely, if $L(\delta\_x)= S$, then $x\in S$. This is clear from the above proof: $\delta\_x\leq L(\delta\_x) = S$.


*Claim 3* : For all $P\in P(X)$, $L(P) = P\cup S$. Indeed, $S= L(0)\leq L(P)$ and $P\leq L(\delta\_x)$. Furthermore, let $z\in L(P)$, i.e. $\delta\_z\leq L(P)$. Then $L(\delta\_z)\leq L(P)$, so that $L(\delta\_z\cap P) = L(\delta\_z)\cap L(P) = L(\delta\_z)$. If $z\notin P$, $\delta\_z\cap P = 0$ and so $L(\delta\_z) = S$ so that by claim 2, $z\in S$. In other words, if $z\in L(P)$, $z\in P$ or $z\in S$.


So any such $L$ is of the form $P\mapsto S\cup P$, and conversely, given any subset $S$, $P\mapsto S\cup P$ is a left exact idempotent monad (in fact, it preserves all limits). So the category of such monads is equivalent to $P(X)$ itself.


I don't know about your dual question, namely about right exact localizations or colocalizations of $Psh(C)$.


