HIGHER DIMENSIONAL NORMALISATION STRATEGIES FOR ACYCLICITY

HIGHER-DIMENSIONAL NORMALISATION STRATEGIES FOR ACYCLICITY YVES GUIRAUD – PHILIPPE MALBOS Abstract – 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 pre- sentation 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. Using track polygraphs, we extend to every dimension the homotopical finiteness condition of finite derivation type, introduced by Squier in string rewriting theory, and we prove that it implies a new homological finiteness condition that we introduce here. The proof is based on normalisation strategies and relates acyclic track polygraphs to free abelian resolutions of the small categories they present. Keywords – rewriting; homology of small categories; low-dimensional topology; identities among relations. M.S.C. 2000 – 18C10; 18D05; 18G10; 18G20; 68Q42. CONTENTS 1 Resolutions by track polygraphs 4 2 Normalisation strategies for track polygraphs 7 3 Track-polygraphic resolutions generated by convergent 2-polygraphs 16 4 Abelianisation of track-polygraphic resolutions 24 5 Examples of track-polygraphic resolutions of small categories 36

• track-polygraphic resolutions


• higher-dimensional track


• reidemeister-fox-squier complex


• string rewriting


• category presented


• rewriting


• category


• fdtp ?


• track