Unification of general relativity and quantum mechanics.

Question

Hello,

This thread concerns a physical notion, but its origins go back to the 

emergence of the theory of mathematical unification. In mathematics, two 

theories are identical, if their corresponding topos (i.e. categories) are 

equivalent.

I would like in this thread to discuss the unification of quantum mechanics, 

and general relativity. What do we mean, mathematically speaking, by unifying 

quantum mechanics and general relativity?

From a few readings that I had done several times in the past, the General 

Theory of Relativity is a geometric theory. On the other hand, quantum 

mechanics is an algebraic theory. So, finally, we seek to unify an algebraic 

theory and a geometric theory. What would that mean?

Classically, I have seen how we unify an algebraic theory and a geometric 

theory, via functors: differential geometry is identified in part with the 

theory of commutative algebra, 

because, the category of real differential varieties, are identified to a 

subcategory of commutative algebras.

Is that, to unify the theory of general relativity, and the theory of quantum 

mechanics consists in exhibiting a functor through which, we identify a 

category of geometric objects within general relativity, and a category 

of algebraic objects within quantum mechanics?

Thanks in advance for your help.