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 $\mathsf{PSh}(\mathcal{C})=\mathsf{Cat}(\mathcal{C}^\mathsf{op},\mathsf{Set})$:

- The 1-categorical analogue of a topological space is a Grothendieck topos, since:
- A Grothendieck topos is a left exact reflective subcategory of $\mathsf{PSh}(\mathcal{C})$;
~~A topology on a set $X$ is a left exact reflective subcategory of the powerset $\mathcal{P}(X)$ of $X$, viewed as a poset;~~(see**Edit**)

- The 1-categorical analogue of a quantale is a "compatibly monoidal cocomplete category", since:
- A quantale is a monoid in suplattices, the algebras for the powerset monad;
- A compatibly monoidal cocomplete category is a pseudomonoid in cocomplete categories, the algebras for the presheaf category relative monad.

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].