Many integrable critical classical statistical mechanical models and the corresponding quantum spin chains possess a fractional-spin conserved current. These currents have been constructed by utilising quantum-group algebras, fermionic and parafermionic operators, and ideas from ``discrete holomorphicity''. I define them generally and naturally using a braided tensor category, a topological structure familiar from the study of knot invariants, anyons and conformal field theory. Such a current amounts to terminating a lattice topological defect, and I will touch on related work on such done with Aasen and Mong. I show how requiring a current be conserved yields simple constraints on the Boltzmann weights, and that all of the many models known to satisfy these constraints are integrable. This procedure therefore gives a linear construction for ``Baxterising'', i.e. building a solution of the Yang-Baxter equation out of topological data.