Let me first give a vague definition of "theory"/"physical theory", see also. A (physical)theory is a collection of rules and notions that were successful in predicting a behaviour of an idealised physical system. This question is about the theories that have stood the test of time.
I would like to know if it is possible to express physical theories, as defined above, in the language of (higher-)categories 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.
Question Is it possible to interpret SR, CM, QM, and GR as objects of some well defined category? This question is only about the physics that are contained in the first 3 volumes of Landau and Lifshitz. However electromagnetism is not considered in this question.