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

$a\times b=c$ in $2+1$D TQFT