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  • Appendix G - Locally Presentable Categories

  • from Appendices
  • 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 601-607

  • Summary

    This appendix is a quick introduction to locally presentable categories. This notion is in some sense a formalization of what is an algebraic structure. When category theory is restricted to locally presentable categories, many things get simpler. In particular, there are characterizations of adjoint functors purely in terms of preservation of limits and colimits. Locally presentable categories also play an important role in the theory of model categories through the concept of combinatorial model categories. There are many ways to define locally presentable categories. The appendix begins by presenting the concept using sketches, which encode the syntax of an algebraic structure. These sketches are used several times in the body of the book. The intrinsic categorical characterization is then provided, introducing several notions that are important for the theory of model categories. Finally, the syntactic characterization is discussed.