Higher-dimensional normalisation strategies for acyclicity

Yves Guiraud 1, 2, 3 Philippe Malbos 1
3 PI.R2 - Design, study and implementation of languages for proofs and programs
PPS - Preuves, Programmes et Systèmes, Inria Paris-Rocquencourt, UPD7 - Université Paris Diderot - Paris 7, CNRS - Centre National de la Recherche Scientifique : UMR7126
Abstract : 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.
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Yves Guiraud, Philippe Malbos. Higher-dimensional normalisation strategies for acyclicity. Advances in Mathematics, Elsevier, 2012, 231 (3-4), pp.2294-2351. ⟨10.1016/j.aim.2012.05.010⟩. ⟨hal-00531242v3⟩

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