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Higher-dimensional normalisation strategies for acyclicity

Authors
Journal
Advances in Mathematics
0001-8708
Publisher
Elsevier
Volume
231
Identifiers
DOI: 10.1016/j.aim.2012.05.010
Keywords
  • Rewriting
  • Polygraphic Resolution
  • Homology Of Small Categories
  • Identities Among Relations
Disciplines
  • Mathematics

Abstract

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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