Decategorifying Grothendieck topoi and categorifying topological spaces

Question

(This is in a sense a follow-up to this question.)

I was under the impression these days 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 $\mathsf{PSh}(\mathcal{C})$?

---

¹Grothendieck topoi are thought as being categorified locales, where the proper categorification of a topological space is an ionad (nLab page, original paper introducing them); see Kevin Arlin's answer here.

Answer 1

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 $Psh(C)$.