06094113

j

06094113 Guiraud, Yves Malbos, Philippe Higher-dimensional normalisation strategies for acyclicity. Adv. Math. 231, No. 3-4, 2294-2351 (2012). 2012 Elsevier Science (Academic Press), San Diego, CA EN rewriting polygraphic resolution homology of small categories identities among relations

doi:10.1016/j.aim.2012.05.010

Summary: 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.