Plato’s cave: Prisoners are chained from birth in an underground cave, able to see nothing but moving shadows cast on the back wall, by a fire in their back. These shadows they take to be the whole of reality. Only after one breaks the chains and turn towards the light, it becomes clear that what used to be taken as reality was nothing but shadow and illusion.
Symmetry can be viewed as the shadow and the illusion of topological order (maybe it really is)
To systematically understand strongly correlated gapless states has been a long standing challenge in theoretical physics. We discover that one of the strongly correlated gapless states, the quantum critical point of Landau symmetry breaking transition in n-dimensional space, has an unbroken algebraic (n-1)-symmetry G(n-1) , in addition to the usual unbroken symmetry G. Symmetry and algebraic higher symmetry together form the larger categorical symmetry, which is denoted as G v G(n-1).
One way to understand categorical symmetry is via non-invertible gravitational anomaly. In fact
categorical symmetry = non-invertible gravitational anomaly = topological order in one higher dimension
In other words, we find that non-invertible gravitational anomaly (i.e. topological order in one higher dimension) is a generic and unified way to describe symmetries, higher symmetries, and the more general algebraic higher symmetries.
For example, the categorical symmetry G v G(n-1) is fully described by the boundary of G-gauge theory in one higher dimension (i.e. in (n+1)-dimensional space). The conservation of the G-gauge charge gives rise to the G symmetry, while the conservation of the G-gauge flux give rise to the G(n-1) algebraic higher symmetry. The gapless boundary of the G-gauge theory has no condensations of the G-gauge charge and the G-gauge flux, and has the unbroken G v G(n-1) categorical symmetry. Such a gapless boundary corresponds to the quantum critical point of G-symmetry breaking transition.
We propose the emergence of categorical symmetry to be a general and essential feature of a gapless quantum liquid, and give examples to show that emergent maximal categorical symmetry determines the low energy dynamics of strongly correlated gapless quantum liquid. We also propose to use the categorical symmetry of a gapless liquid to determine the AdS/CFT dual of the gapless liquid.
Every system with G symmetry is also a system with G v G(n-1) categorical symmetry. Similarly, every system with G(n-1) algebraic higher symmetry is also a system with G v G(n-1) categorical symmetry. However, sometimes, the G(n-1) algebraic higher symmetry is spontaneously broken, and we have an illusion that there is only G symmetry. Other times, the G symmetry is spontaneously broken, and we have an illusion that there is only G(n-1) algebraic higher symmetry. Only in a gapless state, can we have the full unbroken G v G(n-1) categorical symmetry. So G symmetry and G(n-1) algebraic higher symmetry are just different shadows of the same G v G(n-1) categorical symmetry (i.e. different shadows of the G-gauge topological order in one higher dimension).
Armed with such an understanding, we developed higher category theories for algebraic higher symmetry and categorical symmetry, as well as their symmetry enriched topological orders and their symmetry protected trivial orders, based on the above holographic point of view — a view from one higher dimension.
An algebraic higher symmetry is fully described by a local fusion higher category R, which can be viewed as a boundary of a topological order M in one higher dimension.
See:
Algebraic higher symmetry and categorical symmetry — a holographic and entanglement view of symmetry (It took one and half year to finish this paper.)