  We introduce acyclic polygraphs, a notion of complete categorical cellular
model for (small) categories, containing generators, relations and
higher-dimensional globular syzygies. We give a rewriting method to construct
explicit acyclic polygraphs from convergent presentations. For that, we
introduce higher-dimensional normalisation strategies, defined as homotopically
coherent ways to relate each cell of a polygraph to its normal form, then we
prove that acyclicity is equivalent to the existence of a normalisation
strategy. Using acyclic polygraphs, we define a higher-dimensional homotopical
finiteness condition for higher categories which extends Squier's finite
derivation type for monoids. We relate this homotopical property to a new
homological finiteness condition that we introduce here.
