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Topological spaces and locales are two closely related notions meant to capture the concept of an abstract space, the latter of which admits a certain "noncommutative" generalisation known as a quantale. Out of these three notions, topological spaces and quantales admit clear 1-categorical analogues, coming from viewing powersets $\mathcal{P}(X)=\mathsf{Set}(X,\{\mathrm{true},\mathrm{false}\})$ as the 0-categorical analogue of presheaf categories $\mathsf{PSh}(\mathcal{C})=\mathsf{Cat}(\mathcal{C}^\mathsf{op},\mathsf{Set})$:

We have thus assembled the following table:

Topological Spaces Locales Quantales
Grothendieck Topoi ? Compatibly Monoidal Cocomplete Categories

Question 1: What would be the appropriate notion filling the "?" spot in the table above?

It seems to me that one plausible definition of a "2-locale" would be that of a "Cartesian compatibly monoidal cocomplete category", much like locales are quantales whose product is given by the meet (Cartesian product).

Question 2: Has anyone tried developing such a "theory of 2-locales" in this sense?


Edit: I've just realised that a left exact reflective subcategory of $\mathcal{P}(X)$ is actually a different structure (although similar) to that of a topological space. See these two other questions: [1] [2].

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    $\begingroup$ Which def. of a 2-rig do you use? There are (too) many. $\endgroup$ Dec 24, 2022 at 22:46
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    $\begingroup$ @MartinBrandenburg Oh sorry, of course! It's the "compatibly monoidal cocomplete categories" one. By the way, I just noticed I also made a mistake while quoting the result from memory (now corrected): quantales are monoids in suplattices, and these are algebras for the powerset monoid. Similarly algebras for the presheaf relative monad are the cocomplete categories, and pseudomonoids in those are "compatibly monoidal cocomplete categories". $\endgroup$
    – Emily
    Dec 25, 2022 at 2:38

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A standard answer is in fact that Grothendieck toposes are categorified locales, with the argument that a Grothendieck topos that lacks enough points is not a tight generalization of a topological space, noting that the set $X$ of points has no analogue in your analogy between a Grothendieck topos and a space. Seen another way, if you look at the left exact reflective subcategories of presheaves on a poset, i.e. $(0,1)$-categories, then you've got locales.

According to Garner, a categorified topological space should instead by an ionad, which under an accessibility condition is closely related to a Grothendieck topos with enough points. One reason your proposal for a 2-locale might be questioned is that your conditions don't imply any adjoint functor theorem, so that your 2-locales need not be complete, though locales always are.

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