I'd ideally like a categorical definition of differentiability that can then be trivially translated into locales. Barring this, I'm still interested in whether the notion make sense for locales.



Is there a straightforward way to define a differentiable structure on a localic manifold?
This is exactly what I've been working on. I've tried a few approaches. 


Grothendieck gives a definition of formal smoothness which can be translated into just about any category. The problem is that, for the definition to be interesting, it relies on the fact that commutative rings sometimes have non-trivial nilpotents.


G also gives a definition of smoothness which doesn't rely on nilpotents necessarily. This is what I've been working on. It's possible to give the definition, but it's not immediately clear whether it's interesting in the sense that there are some arrows in Loc which are smooth and some which are not. There is some promise, though, in that, smooth arrows are flat and not everything in the category of locales is flat.


What got you interested in this by the way? Feel free to email me. 


