Note on lattice description of generalized symmetries in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> gauge theories

Topology and generalized symmetries in the $SU(N)/{\mathbb{Z}}_{N}$ gauge theory are considered continuum lattice. Starting from $SU(N)$ with 't Hooft twisted boundary condition, we give a simpler explanation of van Baal's proof on fractionality topological charge. This description is applicable to both lattice by using L\"uscher's construction topology Thus can recover principal bundle fields being subject ${\mathbb{Z}}_{N}$-relaxed cocycle condition. We explicitly demonstrate fractional charge, verify an equivalence other constructions reported recently based different ideas. Gauging ${\mathbb{Z}}_{N}$ 1-form center symmetry enables theories couple 2-form field as simple integer field, reproduce Kapustin-Seiberg prescription limit. Our also applied analyzing higher-group structure instanton-sum modification.