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question: What are the mathematical theories suitable to describe the "continuous Phase transitions between Category Theories"? The phase transitions mean that in terms of the quantum statistical physics, in the sense as the second order or higher order continuous phase transitions between two distinct phases.

Category Theories can be useful to describe the intrinsic topologically ordered quantum matter. The bulk part phases $A$ and $B$ of the 2+1 spacetime dimensional topological order can be constructed using a unitary tensor category $\mathcal{C}$, denoted $\mathcal{C}_A$ and $\mathcal{C}_B$. The boundary of the topological order described by tensor category $\mathcal{C}$ is associated with a module category over $\mathcal{C}$. See for example, the work of Kong-Kitaev.

One can also consider domain walls (or defect lines) between different bulk topological orders, between $\mathcal{C}_A$ and $\mathcal{C}_B$. If the bulk topological orders $\mathcal{C}_A$ and $\mathcal{C}_B$ are the unitary modular tensor categories, then the gapped domain walls can be regarded as the bimodule categories between modular tensor categories. One may use the fusion space dimension $\mathcal{W}_{ab}$ that tunneling between two topological orders $\mathcal{C}_A$ and $\mathcal{C}_B$, to label different types of gapped domain walls. Here $a$ and $b$ are the objects (the anyon or quasiparticle contents) of the unitary modular tensor categories theories. For example, the work of Lan-Wang-Wen and Kawahigashi, and the work of Rehren (the coupling matrix), Davydov, Müger, Nikshych and Ostrik.

However, the domain walls (or defect lines) between different bulk topological orders are the spatial junction between different bulk topological orders, or the spatial junction between different Category Theories. Do we have methods to describe the temporal junction between different bulk topological orders, say along the phase transitions along the time direction by tuning some coupling parameters in the quantum Hamiltonian? Namely, the temporal junction between different two (unitary modular tensor) category $\mathcal{C}_A$ and $\mathcal{C}_B$? This is the (temporal) phase transition between the two Category Theories. What are the mathematical theories suitable to describe the "(temporal) Phase transitions between Category Theories"? Such that the (temporal) Phase transitions is continuous and smooth? As the second order or higher order continuous phase transitions in terms of the statistical physics.




Other information:

This table bridges the simple terminology between physical excitations and Category Theories

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This table bridges the terminology between topological orders and Category Theories: enter image description here

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    $\begingroup$ All of the bold makes it difficult to read. $\endgroup$ – user13113 Dec 19 '16 at 0:51
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In general, we expect field theories to be described by some higher categorical structures, where bulk models are assigned objects (also called 0-morphisms), domain walls are assigned morphisms (also called 1-morphisms), and higher-codimension defects are assigned higher morphisms. For the example of Levin-Wen models, there is a 3-category of tensor categories, where the morphisms are bimodule categories, and 2-morphisms are functors between bimodule categories. In this context, there does not seem to be a substantial difference between spacial domain walls and temporal domain walls.

If you want a continuously parametrized phase change, then you may want to consider simplicial maps from the singular complex of the interval to the $(\infty,1)$-category of "tensor categories" (or whatever your bulk structures happen to be). For Levin-Wen, each point in the interval is assigned a tensor category, and there are compatible morphisms (i.e., bimodules) between each pair of points. More generally, you may replace the interval with your spacetime manifold to get a fully varying family of tensor categories. Smoothness requires some infinitesimal structure that I do not know how to derive.

I do not know of a reference for your precise question, but I think Gaitsgory-Lurie's work on chiral categories has some information on families or local systems of $(\infty,1)$-categories. There is also a brief treatment of "constructible sheaves" in an appendix of Lurie's "Higher Algebra", but the topos language may be difficult to translate.

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