Let me first give a vague definition of "theory"/"physical theory", [see also](https://ncatlab.org/nlab/show/theory+%28physics%29). 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](https://arxiv.org/abs/1310.7930v1), 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.


**EDIT :** I would like to add two links to physics.stackexchange. I really liked Schreiber's answer to [this question](https://physics.stackexchange.com/questions/246208/implementing-category-theory-in-general-relativity/246280#246280) as well as to [this question](https://physics.stackexchange.com/questions/94560/hilbert-g%C3%B6del-and-god-equations-a-19th-century-lesson-for-21st-century-phy/97883#97883).



Mathematical formalization of physics
[John Baez](https://arxiv.org/abs/quant-ph/0404040v2) (2004) discusses quantum theory and general relativity from a unified perspective provided by category theory.



> 
> Faced with the great challenge of reconciling general relativity and
> quantum theory, it is difficult to know just how deeply we need to
> rethink basic concepts. Here we propose that quantum theory will make
> more sense when regarded as part of a theory of spacetime. This calls
> for a category-theoretic perspective — in particular, one that
> de-emphasizes the primary role of the category of sets and functions, relevant for classical physics.
> 
> 
> The category
> **Hilb**, whose objects are Hilbert spaces and whose morphisms are linear
> operators between these, plays an important role in quantum
> theory. The category $n$**Cob**, whose objects are ($n$− 1)-dimensional manifolds and whose morphisms are $n$-dimensional
> manifolds, plays an important in general relativity.
> 
> 
> While an individual manifold is not very much like a Hilbert space,
> the category $n$**Cob** turns out to have many structural similarities to
> the category **Hilb**. The most puzzling features of quantum
> theory all arise from ways in which **Hilb** resembles $n$**Cob** more than
> the "classical physics" category **Set**, whose objects are sets and whose morphisms are
> functions.
> 
> 
> 


