Accessible category

The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects.

The theory originates in the work of Grothendieck completed by 1969,[1] and Gabriel and Ulmer (1971).[2] It has been further developed in 1989 by Michael Makkai and Robert Paré, with motivation coming from model theory, a branch of mathematical logic.[3] A standard text book by Adámek and Rosický appeared in 1994.[4] Accessible categories also have applications in homotopy theory.[5][6] Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs.[7] Some properties of accessible categories depend on the set universe in use, particularly on the cardinal properties and Vopěnka's principle.[8]

  1. ^ Grothendieck, Alexander; et al. (1972), Théorie des Topos et Cohomologie Étale des Schémas, Lecture Notes in Mathematics 269, Springer
  2. ^ Gabriel, P; Ulmer, F (1971), Lokal Präsentierbare Kategorien, Lecture Notes in Mathematics 221, Springer
  3. ^ Makkai, Michael; Paré, Robert (1989), Accessible categories: The foundation of Categorical Model Theory, Contemporary Mathematics, AMS, ISBN 0-8218-5111-X
  4. ^ Adámek, Jiří; Rosický, Jiří (10 March 1994). Locally Presentable and Accessible Categories. Cambridge University Press. doi:10.1017/cbo9780511600579. ISBN 978-0-521-42261-1.
  5. ^ J. Rosický "On combinatorial model categories", arXiv, 16 August 2007. Retrieved on 19 January 2008.
  6. ^ Rosický, J. "Injectivity and accessible categories." Cubo Matem. Educ 4 (2002): 201-211.
  7. ^ Grothendieck, Alexander (1991), Les dérivateurs, Contemporary Mathematics, manuscript (Les Dérivateurs: Texte d'Alexandre Grothendieck. Édité par M. Künzer, J. Malgoire, G. Maltsiniotis)
  8. ^ Adamek/Rosický 1994, chapter 6