(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:
- Preserving colimits, i.e. $L(\emptyset)=\emptyset$ and $L\big(\bigcup_{i\in I}U_i\big)=\bigcup_{i\in I}L(U_i)$;
- 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.