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1 results

  • 22 - Homology of ω-Categories

  • 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 440-451

  • Summary

    This chapter is about Métayer’s polygraphic homology of ω-categories. This homology theory was first defined in the following way: the polygraphic homology of an ω-category is the homology of the abelianization of any of its polygraphic replacements. Métayer then showed with Lafont that for every monoid, considered as an ω-category, its polygraphic homology coincides with its classical homology as a monoid. This result was then generalized to 1-categories by Guetta. In this chapter, it is proven that the polygraphic homology is the left derived functor of a linearization functor from the category of ω-categories to the category of chain complexes, respectively endowed with ω-equivalences and quasi-isomorphisms.