Modular nondegeneracy

Let $\mathcal{C}$ be a unitary ribbon fusion category. An object $x$ is transparent if for every simple object $y$ the double braiding

$c_{y,x}\circ c_{x,y}:x\otimes y⟶x\otimes y$

is the identity. By naturality and additivity, the double braiding on a finite direct sum is block diagonal with the double braidings on its simple summands; therefore an object is transparent exactly when each simple summand is transparent. Finite direct sums include the empty sum, so the zero object is transparent. The category is modular, or nondegenerate, when its transparent simple objects consist only of the tensor unit, equivalently when its transparent objects are exactly the finite direct sums of the tensor unit. This condition is additional to the existence of fusion spaces and a unitary braid action.