We introduce acyclic track polygraphs, a notion of complete categorical cellular models for small categories: they are polygraphs containing generators, with additional invertible cells for relations and higher-dimensional globular syzygies. We give a rewriting method to realise such a model by proving that a convergent presentation canonically extends to an acyclic track polygraph. For that, we introduce normalising strategies, defined as homotopically coherent ways to relate each cell of a track polygraph to its normal form, and we prove that acyclicity is equivalent to the existence of a normalisation strategy.