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Journal Articles Advances in Mathematics Year : 2012

Higher-dimensional normalisation strategies for acyclicity

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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Dates and versions

hal-00531242 , version 1 (02-11-2010)
hal-00531242 , version 2 (13-06-2012)
hal-00531242 , version 3 (08-08-2012)

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Cite

Yves Guiraud, Philippe Malbos. Higher-dimensional normalisation strategies for acyclicity. Advances in Mathematics, 2012, 231 (3-4), pp.2294-2351. ⟨10.1016/j.aim.2012.05.010⟩. ⟨hal-00531242v3⟩
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