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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.

  • A 2-categorical proof of Frobenius for fibrations defined from a generic point

  • Sina Hazratpour, Emily Riehl
  • Journal:
    Mathematical Structures in Computer Science / Volume 34 / Issue 4 / April 2024
    Published online by Cambridge University Press:
    15 April 2024, pp. 258-280
    • Article
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  • Consider a locally cartesian closed category with an object $\mathbb{I}$ and a class of trivial fibrations, which admit sections and are stable under pushforward and retract as arrows. Define the fibrations to be those maps whose Leibniz exponential with the generic point of $\mathbb{I}$ defines a trivial fibration. Then the fibrations are also closed under pushforward.