  A braid representation is a monoidal functor from the braid category
$\mathsf{B}$, for example given by a solution to the constant Yang-Baxter
equation. Given a monoidal category $\mathsf{C}$ with
$ob(\mathsf{C})=\mathbb{N}$, a rank-$N$ charge-conserving representation (or
spin-chain representation) is a strict monoidal functor $F$ from $\mathsf{C}$
to the category $\mathrm{Match}^N$ of rank-$N$ charge-conserving matrices that
is natural in the sense that $F(1)=1$}. In this work we construct all
spin-chain braid representations, and classify up to suitable notions of
isomorphism.
