I would like to know if it is possible to use the language of (higher-)categories to formalize all the known theories in physics and then to consider the "category of all the theories". Since I am not an expert this question will be quite vague and speculative. It looks that the book Differential cohomology in a cohesive infinity-topos, gives a partial answer to my question, but it is quite long and sometimes difficult to read. Would you kindly provide me with some shorter references?

Very naively, when you "take the limit $c \to +\infty$" in Special Relativity(SR) you recover Classical Mechanics(CM) and when you "take the limit $\hbar \to 0$" in Quantum Mechanics(QM) you also recover Classical Mechanics. In the same fashion by considering flat metric in General Relativity(GR) you recover Special Relativity. It appears that, in an appropriate setting (if such a setting would exist), that SR is a unique solution to a deformation problem and there is a canonical arrow SR $\to$ CM, and similarly for QM $\to$ CM and GR $\to$ SR. Probably the Quantum Gravity could also be interpreted as a solution to some deformation problem. A possible candidate is a fibre product of GR and QM. The hope is to use some properties of the "category of all the theories" to prove some abstract existence results in a more general setting.

Question Is it possible to interpret SR, CM, QM, and GR as objects of some well defined category?