What do Abelian categories form?
- Authors: Kaledin D.B.1,2
-
Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Laboratory of algebraic geometry and its applications, National Research University "Higher School of Economics" (HSE)
- Issue: Vol 77, No 1 (2022)
- Pages: 3-54
- Section: Articles
- URL: https://journal-vniispk.ru/0042-1316/article/view/133687
- DOI: https://doi.org/10.4213/rm10044
- ID: 133687
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Abstract
Given two finitely presentable Abelian categories $A$ and $B$, we outline a construction of an Abelian category of functors from $A$ to $B$, which has nice 2-categorical properties and provides an explicit model for a stable category of stable functors between the derived categories of $A$ and $B$. The construction is absolute, so it makes it possible to recover not only Hochschild cohomology but also Mac Lane cohomology.Bibliography: 29 titles.
About the authors
Dmitry Borisovich Kaledin
Steklov Mathematical Institute of Russian Academy of Sciences; Laboratory of algebraic geometry and its applications, National Research University "Higher School of Economics" (HSE)
Email: kaledin@mi-ras.ru
Doctor of physico-mathematical sciences, no status
References
- M. Artin, A. Grothendieck, J. L. Verdier (eds.), Theorie de topos et cohomologie etale des schemas, Seminaire de geometrie algebrique du Bois-Marie 1963–1964 (SGA 4). Dirige par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne, B. Saint-Donat, v. 1, Lecture Notes in Math., 269, Springer-Verlag, Berlin–New York, 1972, xix+525 pp.
- A. A. Beilinson, J. Bernstein, P. Deligne, “Faisceaux pervers”, Analysis and topology on singular spaces (Luminy, 1981), v. I, Asterisque, 100, Soc. Math. France, Paris, 1982, 5–171
- A. K. Bousfield, “Constructions of factorization systems in categories”, J. Pure Appl. Algebra, 9:2 (1976/77), 207–220
- И. Букур, А. Деляну, Введение в теорию категорий и функторов, Мир, М., 1972, 259 с.
- P. Deligne, “Theorie de Hodge. III”, Inst. Hautes Etudes Sci. Publ. Math., 44 (1974), 5–77
- A. Dold, “Homology of symmetric products and other functors of complexes”, Ann. of Math. (2), 68 (1958), 54–80
- W. G. Dwyer, P. S. Hirschhorn, D. M. Kan, J. H. Smith, Homotopy limit functors on model categories and homotopical categories, Math. Surveys Monogr., 113, Amer. Math. Soc., Providence, RI, 2004, viii+181 pp.
- W. G. Dwyer, J. Spalinski, “Homotopy theories and model categories”, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, 73–126
- С. И. Гельфанд, Ю. И. Манин, Методы гомологической алгебры, т. I, Наука, М., 1988, 416 с.
- А. Гротендик, О некоторых вопросах гомологической алгебры, Библиотека сборника “Математика”, ИЛ, М., 1961, 175 с.
- 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
- M. Hovey, Model categories, Math. Surveys Monogr., 63, Amer. Math. Soc., Providence, RI, 1999, xii+209 pp.
- M. Jibladze, T. Pirashvili, “Cohomology of algebraic theories”, J. Algebra, 137:2 (1991), 253–296
- П. Т. Джонстон, Теория топосов, Наука, М., 1986, 439 с.
- D. Kaledin, “Trace theories and localization”, Stacks and categories in geometry, topology, and algebra, Contemp. Math., 643, Amer. Math. Soc., Providence, RI, 2015, 227–262
- D. Kaledin, “How to glue derived categories”, Bull. Math. Sci., 8:3 (2018), 477–602
- Д. Б. Каледин, “Сопряженность в 2-категориях”, УМН, 75:5(455) (2020), 101–152
- D. Kaledin, W. Lowen, “Cohomology of exact categories and (non-)additive sheaves”, Adv. Math., 272 (2015), 652–698
- M. Kashiwara, P. Schapira, Categories and sheaves, Grundlehren Math. Wiss., 332, Springer-Verlag, Berlin, 2006, x+497 pp.
- B. Keller, “On differential graded categories”, International congress of mathematicians, v. II, Eur. Math. Soc., Zürich, 2006, 151–190
- M. Kontsevich, Y. Soibelman, “Notes on $A_infty$-algebras, $A_infty$-categories and non-commutative geometry”, Homological mirror symmetry, Lecture Notes in Phys., 757, Springer, Berlin, 2009, 153–219
- W. Lowen, M. Van den Bergh, “Deformation theory of abelian categories”, Trans. Amer. Math. Soc., 358:12 (2006), 5441–5483
- D. G. Quillen, Homotopical algebra, Lecture Notes in Math., 43, Springer-Verlag, Berlin–New York, 1967, iv+156 pp.
- D. Quillen, “Higher algebraic K-theory. I”, Algebraic K-theory (Battelle Memorial Inst., Seattle, WA, 1972), v. I, Lecture Notes in Math., 341, Higher K-theories, Springer, Berlin, 1973, 85–147
- D. Schäppi, “Ind-abelian categories and quasi-coherent sheaves”, Math. Proc. Cambridge Philos. Soc., 157:3 (2014), 391–423
- D. Tamarkin, “What do dg-categories form?”, Compos. Math., 143:5 (2007), 1335–1358
- J. L. Verdier, “Topologies et faisceaux”, Exp. II, 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, Springer-Verlag, Berlin–New York, 1972, 219–263
- 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.
- Ch. A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math., 38, Cambridge Univ. Press, Cambridge, 1994, xiv+450 pp.
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