  Many integrable statistical mechanical models possess a fractional-spin
conserved current. Such currents have been constructed by utilising
quantum-group algebras and ideas from "discrete holomorphicity". I find them
naturally and much more generally using a braided tensor category, a
topological structure arising in knot invariants, anyons and conformal field
theory. I derive a simple constraint on the Boltzmann weights admitting a
conserved current, generalising one found using quantum-group algebras. The
resulting trigonometric weights are typically those of a critical integrable
lattice model, so the method here gives a linear way of "Baxterising", i.e.
building a solution of the Yang-Baxter equation out of topological data. It
also illuminates why many models do not admit a solution. I discuss many
examples in geometric and local models, including (perhaps) a new solution.
