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2 results

  • 19 - Polygraphic Resolutions

  • from Part V - Homotopy Theory of Polygraphs
  • 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 381-391

  • Summary

    The purpose of this chapter is to introduce the notion of a polygraphic resolution of an ω-category. This notion was introduced by Métayer to define a homology theory for ω-categories, that is now known as the polygraphic homology. It was then showed by himself and Lafont that this homology recovers the classical homology of monoids for ω-categories coming from monoids. It is now known by work of Lafont, Métayer, and Worytkiewicz that these polygraphic resolutions are resolutions in the sense of a model category structure on ω-categories, the so-called folk model structure. Every ω-category is shown to admit such a resolution, and the relationship between two resolutions of the same ω-category is examined.

  • Quillen model categories

  • Mark Hovey
  • Journal:
    Journal of K-Theory / Volume 11 / Issue 3 / June 2013
    Published online by Cambridge University Press:
    04 March 2013, pp. 469-478
    Print publication:
    June 2013

  • We provide a brief description of the mathematics that led to Daniel Quillen's introduction of model categories, a summary of his seminal work “Homotopical algebra”, and a brief description of some of the developments in the field since.