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For a zero-temperature Landau symmetry breaking transition in $n$-dimensionalspace that completely breaks a finite symmetry $G$, the critical point at thetransition has the symmetry $G$. In this paper, we show that the critical pointalso has a dual symmetry - a $(n-1)$-symmetry described by a higher group when$G$ is Abelian or an algebraic $(n-1)$-symmetry beyond higher group when $G$ isnon-Abelian. In fact, any $G$-symmetric system can be viewed as a boundary of$G$-gauge theory in one higher dimension. The conservation of gauge charge andgauge flux in the bulk $G$-gauge theory gives rise to the symmetry and the dualsymmetry respectively. So any $G$-symmetric system actually has a largersymmetry called categorical symmetry, which is a combination of the symmetryand the dual symmetry. However, part (and only part) of the categoricalsymmetry must be spontaneously broken in any gapped phase of the system, butthere exists a gapless state where the categorical symmetry is notspontaneously broken. Such a gapless state corresponds to the usual criticalpoint of Landau symmetry breaking transition. The above results remain valideven if we expand the notion of symmetry to include higher symmetries andalgebraic higher symmetries. Thus our result also applies to critical pointsfor transitions between topological phases of matter. In particular, we showthat there can be several critical points for the transition from the 3+1D$Z_2$ gauge theory to a trivial phase. The critical point from Higgscondensation has a categorical symmetry formed by a $Z_2$ 0-symmetry and itsdual - a $Z_2$ 2-symmetry, while the critical point of the confinementtransition has a categorical symmetry formed by a $Z_2$ 1-symmetry and its dual- another $Z_2$ 1-symmetry.

Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions