  We study the implications of the anyon fusion equation $a\times b=c$ on
global 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 is
well known, when at least one of $a$ and $b$ is abelian, such equations
describe aspects of the one-form symmetry of the theory. When $a$ and $b$ are
non-abelian, the most obvious way such fusions arise is when a TQFT can be
resolved 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 such
fusions for non-abelian $a$ and $b$ can also be an indication of zero-form
symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the
case of discrete gauge theories based on the largest Mathieu group, $M_{24}$),
or of the existence of non-modular fusion subcategories. We study these ideas
in a variety of TQFT settings from (twisted and untwisted) discrete gauge
theories to Chern-Simons theories based on continuous gauge groups and related
cosets. Along the way, we prove various useful theorems.
