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 $\mathsf{PSh}(\mathcal{C})$](https://ncatlab.org/nlab/show/sheaf+toposes+are+equivalently+the+left+exact+reflective+subcategories+of+presheaf+toposes);
	+ ~~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](https://ncatlab.org/nlab/show/2-rig#MonoidalCompleteCateories)", 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?




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**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]](https://mathoverflow.net/questions/342466) [[2]](https://mathoverflow.net/questions/437484).



Is there a good theory of 2-locales?
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,](https://ncatlab.org/nlab/show/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.


