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1 results

  • 7 - Coherence by Convergence

  • from Part II - Coherent Presentations
  • Dimitri Ara, Aix-Marseille Université, Albert Burroni, Université Paris Cité, Yves Guiraud, Université Paris Cité, Philippe Malbos, Université Claude Bernard Lyon 1, François Métayer, Université Paris Cité, Samuel Mimram, École Polytechnique, Paris
  • Book:
    Polygraphs: From Rewriting to Higher Categories
    Published online:
    18 March 2025
    Print publication:
    03 April 2025, pp 149-179

  • Summary

    This chapter introduces the notion of acyclic extension of a 2-category, which consists of the additional data of 3-generators "filling all the spheres". This leads to the notion of coherent presentation of a category, which consists of a 2-polygraph presenting the category together with an acyclic extension of the free (2,1)-category on the polygraph. Coherent presentations are then constructed from convergent ones, and the appropriate notion of Tietze transformation between coherent presentations is studied: this allows formulation of a coherent variant of the Knuth-Bendix completion procedure, but also a reduction procedure, which can be used to obtain smaller coherent presentations. Finally, coherent presentations of algebras are studied, thereby defining the proper notion of coherent extension for linear polygraphs.