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.

Final version

References in corpus (5)

• Termination orders for 3-dimensional rewriting
• Higher-dimensional categories with finite derivation type
• The Brown-Golasinski model structure on strict $\infty$-groupoids revisited
• Identities among relations for higher-dimensional rewriting systems
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Cited by in corpus (16)

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• Coherent presentations of Artin monoids
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• A Quillen's Theorem A for strict $\infty$-categories I: the simplicial proof
• Finite convergent presentation of plactic monoid for type C
• A coherence theorem for pseudonatural transformations
• Topological finiteness properties of monoids. Part 1: Foundations
• Algebraic coherent confluence and higher globular Kleene algebras
• Chinese syzygies by insertions
• Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensions
• Coherence in monoidal track categories
• Topological closure of formal power series ideals and application to topological rewriting theory
• Freely generated $n$-categories, coinserters and presentations of low dimensional categories
• Zigzag normalisation for associative $n$-categories
• Higher Catoids, Higher Quantales and their Correspondences
• Strategies for linear rewriting systems: link with parallel rewriting and involutive divisions