<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi><mml:mo>×</mml:mo><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi></mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>D TQFT

We study the implications of anyon fusion equation $a\times b=c$ on global properties $2+1$D topological quantum field theories (TQFTs). Here $a$ and $b$ are anyons that fuse together to give a unique anyon, $c$. As is well known, when at least one abelian, such equations describe aspects one-form symmetry theory. When non-abelian, most obvious way fusions arise TQFT can be resolved into product TQFTs with trivial mutual braiding, lie in separate factors. More generally, we argue appearance for non-abelian also an indication zero-form symmetries TQFT, what term "quasi-zero-form symmetries" (as case discrete gauge based largest Mathieu group, $M_{24}$), or existence non-modular subcategories. these ideas variety settings from (twisted untwisted) Chern-Simons continuous groups related cosets. Along way, prove various useful theorems.