  Inspired by quantum information theory, we look for representations of the
braid groups $B_n$ on $V^{\otimes (n+m-2)}$ for some fixed vector space $V$
such that each braid generator $\sigma_i, i=1,...,n-1,$ acts on $m$ consecutive
tensor factors from $i$ through $i+m-1$. The braid relation for $m=2$ is
essentially the Yang-Baxter equation, and the cases for $m>2$ are called
generalized Yang-Baxter equations. We observe that certain objects in ribbon
fusion categories naturally give rise to such representations for the case
$m=3$. Examples are given from the Ising theory (or the closely related
$SU(2)_2$), $SO(N)_2$ for $N$ odd, and $SU(3)_3$. The solution from the
Jones-Kauffman theory at a $6^{th}$ root of unity, which is closely related to
$SO(3)_2$ or $SU(2)_4$, is explicitly described in the end.
