How to enhance categories, and why?

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Abstract

This is an overview of an approach to the theory of homotopically enhanced categories based on Grothendieck's idea of a ‘derivator’.Bibliography: 27 titles.

About the authors

Dmitry Borisovich Kaledin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia; National Research University "Higher School of Economics", Moscow, Russia

Email: kaledin@mi-ras.ru
Doctor of physico-mathematical sciences, no status

References

  1. J. Adamek, J. Rosicky, Locally presentable and accessible categories, London Math. Soc. Lecture Note Ser., 189, Cambridge Univ. Press, Cambridge, 1994, xiv+316 pp.
  2. M. Artin, A. Grothendieck, J. L. Verdier, N. Bourbaki, P. Deligne, B. Saint-Donat (eds.), Theorie de topos et cohomologie etale des schemas, Seminaire de geometrie algebrique du Bois-Marie 1963–1964 (SGA 4), v. 1, Lecture Notes in Math., 269, Theorie des topos, Exp. I–IV, Springer-Verlag, Berlin–New York, 1972, xix+525 pp.
  3. M. Artin, A. Grothendieck, J. L. Verdier, P. Deligne, B. Saint-Donat (eds.), Theorie de topos et cohomologie etale des schemas, Seminaire de geometrie algebrique du Bois-Marie 1963–1964 (SGA 4), v. 2, Lecture Notes in Math., 270, Exp. V–VIII, Springer-Verlag, Berlin–New York, 1972, iv+418 pp.
  4. M. Artin, A. Grothendieck, J. L. Verdier, P. Deligne, B. Saint-Donat (eds.), Theorie de topos et cohomologie etale des schemas, Seminaire de geometrie algebrique du Bois-Marie 1963–1964 (SGA 4), v. 3, Lecture Notes in Math., 305, Exp. IX–XIX, Springer-Verlag, Berlin–New York, 1973, vi+640 pp.
  5. C. Barwick, D. M. Kan, “Relative categories: another model for the homotopy theory of homotopy theories”, Indag. Math. (N. S.), 23:1-2 (2012), 42–68
  6. A. K. Bousfield, “Constructions of factorization systems in categories”, J. Pure Appl. Algebra, 9:2-3 (1976/77), 207–220
  7. E. H. Brown, Jr., “Cohomology theories”, Ann. of Math. (2), 75:3 (1962), 467–484
  8. P. Gabriel, F. Ulmer, Lokal präsentierbare Kategorien, Lecture Notes in Math., 221, Springer-Verlag, Berlin–New York, 1971, v+200 pp.
  9. M. Groth, “Derivators, pointed derivators and stable derivators”, Algebr. Geom. Topol., 13:1 (2013), 313–374
  10. A. Grothendieck, “Sur quelques points d'algèbre homologique”, Tohoku Math. J. (2), 9 (1957), 119–221
  11. A. Grothendieck, “Categories fibrees et descente”, Revêtements etales et groupe fondamental, Seminaire de geometrie algebrique du Bois Marie 1960/1961 (SGA 1), Lecture Notes in Math., 224, Springer-Verlag, Berlin–New York, 1971, Exp. VI, 145–194
  12. A. Grothendieck, À la poursuite des champs, unpublished manuscript, 1983
  13. A. Grothendieck, Esquisse d'un programme, unpublished manuscript, 1984
  14. M. Hovey, Model categories, Math. Surveys Monogr., 63, Amer. Math. Soc., Providence, RI, 1999, xii+209 pp.
  15. П. Т. Джонстон, Теория топосов, Наука, М., 1986, 439 с.
  16. D. Kaledin, “How to glue derived categories”, Bull. Math. Sci., 8:3 (2018), 477–602
  17. Д. Б. Каледин, “Сопряженность в 2-категориях”, УМН, 75:5(455) (2020), 101–152
  18. D. Kaledin, Enhancement for categories and homotopical algebra, 2024, 595 pp.
  19. D. Kaledin, “Taming large categories”, São Paulo J. Math. Sci., 18:2 (2024), 773–800
  20. J. Lurie, Higher topos theory, Ann. of Math. Stud., 170, Princeton Univ. Press, Princeton, NJ, 2009, xviii+925 pp.
  21. M. Makkai, R. Pare, Accessible categories: the foundation of categorical model theory, Contemp. Math., 104, Amer. Math. Soc., Providence, RI, 1989, viii+176 pp.
  22. D. G. Quillen, Homotopical algebra, Lecture Notes in Math., 43, Springer-Verlag, Berlin–New York, 1967, iv+156 pp.
  23. C. L. Reedy, Homology of algebraic theories, Ph.D. thesis, Univ. of California, San Diego, 1974, 52 pp.
  24. C. Rezk, “A model for the homotopy theory of homotopy theory”, Trans. Amer. Math. Soc., 353:3 (2001), 973–1007
  25. Р. М. Свитцер, Алгебраическая топология – гомотопии и гомологии, Наука, М., 1985, 607 с.
  26. B. Toën, “Vers une axiomatisation de la theorie des categories superieures”, $K$-Theory, 34:3 (2005), 233–263
  27. J.-L. Verdier, Des categories derivees des categories abeliennes, With a preface by L. Illusie, edited and with a note by G. Maltsiniotis, Asterisque, 239, Soc. Math. France, Paris, 1996, xii+253 pp.

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