What do abelian categories form? (English. Russian original) Zbl 1495.18011
Russ. Math. Surv. 77, No. 1, 1-45 (2022); translation from Usp. Mat. Nauk 77, No. 1, 3-54 (2022).
This paper deals with the construction of a nicely behaved abelian category of functors \(\operatorname{Mor}(\mathcal A, \mathcal B)\) between finitely presented abelian categories \(\mathcal A\) and \(\mathcal B\). This also yields a suitable derived category of stable functors between \(\mathcal A\) and \(\mathcal B\), from which one obtains triangulated functors between the (bounded below) derived categories of \(\mathcal A\) and \(\mathcal B\). Moreover, this construction is absolute, i.e. it does not depend on the choice of a fixed ground field. For this reason, it can be used to recover Mac Lane cohomology.
Given finitely presented abelian categories \(\mathcal A\) and \(\mathcal B\), the Grothendieck abelian category \(\operatorname{Mor}(\mathcal A,\mathcal B)\) is defined as a category of sheaves on a suitable topology. Its objects are not (in general) additive functors. By specializing to its full subcategory \(\operatorname{Mor}_{\textrm{add}}(\mathcal A, \mathcal B)\) of additive functors, one recovers the Grothendieck abelian category of left exact functors \(\mathcal A \to \mathcal B\) which preserve filtered colimits. In particular, if \(\mathcal C\) is a small abelian category and \(\mathrm{Ab}\) is the abelian category of abelian groups, it can be shown that \(\operatorname{Mor}_{\textrm{add}}(\operatorname{Ind}(\mathcal C), \mathrm{Ab})\) is equivalent to \(\operatorname{Ind}(\mathcal C)\) itself, the inductive completion of \(\mathcal C\).
Next, the derived category \(\operatorname{DMor}(\mathcal A, \mathcal B)\) of \(\operatorname{Mor}(\mathcal A,\mathcal B)\) is defined. Not every functor in \(\operatorname{DMor}(\mathcal A, \mathcal B)\) is stable — which roughly means that it preserves homotopy bicartesian squares — and we can take the full subcategory \(\operatorname{DMor}_{\mathrm{st}}(\mathcal A, \mathcal B)\) of stable objects. This category has a t-structure whose heart is the category \(\operatorname{Mor}_{\textrm{add}}(\mathcal A, \mathcal B)\), but it is not equivalent to the derived category of such heart. From this, one can recover Mac Lane cohomology classes.
Given finitely presented abelian categories \(\mathcal A\) and \(\mathcal B\), the Grothendieck abelian category \(\operatorname{Mor}(\mathcal A,\mathcal B)\) is defined as a category of sheaves on a suitable topology. Its objects are not (in general) additive functors. By specializing to its full subcategory \(\operatorname{Mor}_{\textrm{add}}(\mathcal A, \mathcal B)\) of additive functors, one recovers the Grothendieck abelian category of left exact functors \(\mathcal A \to \mathcal B\) which preserve filtered colimits. In particular, if \(\mathcal C\) is a small abelian category and \(\mathrm{Ab}\) is the abelian category of abelian groups, it can be shown that \(\operatorname{Mor}_{\textrm{add}}(\operatorname{Ind}(\mathcal C), \mathrm{Ab})\) is equivalent to \(\operatorname{Ind}(\mathcal C)\) itself, the inductive completion of \(\mathcal C\).
Next, the derived category \(\operatorname{DMor}(\mathcal A, \mathcal B)\) of \(\operatorname{Mor}(\mathcal A,\mathcal B)\) is defined. Not every functor in \(\operatorname{DMor}(\mathcal A, \mathcal B)\) is stable — which roughly means that it preserves homotopy bicartesian squares — and we can take the full subcategory \(\operatorname{DMor}_{\mathrm{st}}(\mathcal A, \mathcal B)\) of stable objects. This category has a t-structure whose heart is the category \(\operatorname{Mor}_{\textrm{add}}(\mathcal A, \mathcal B)\), but it is not equivalent to the derived category of such heart. From this, one can recover Mac Lane cohomology classes.
Reviewer: Francesco Genovese (Praha)
MSC:
| 18E10 | Abelian categories, Grothendieck categories |
| 18N10 | 2-categories, bicategories, double categories |
References:
| [1] | Artin, M.; Grothendieck, A.; ; J. L. Verdier (eds.); Artin, M.; Grothendieck, A.; ; J. L. Verdier (eds.); Artin, M.; Grothendieck, A.; ; J. L. Verdier (eds.), Théorie de topos et cohomologie étale des schémas. Vol. 1, Séminaire de géométrie algébrique du Bois-Marie 1963-1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne, B. Saint-Donat. Théorie de topos et cohomologie étale des schémas. Vol. 2. Théorie de topos et cohomologie étale des schémas. Vol. 3, 305, vi+640 pp. (1973), Springer-Verlag: Springer-Verlag, Berlin-New York · Zbl 0245.00002 · doi:10.1007/BFb0070714 |
| [2] | Beilinson, A. A.; Bernstein, J.; ; Deligne, P., Faisceaux pervers, Analysis and topology on singular spaces. Vol. I, 100, 5-171 (1982), Soc. Math. France: Soc. Math. France, Paris · Zbl 0536.14011 |
| [3] | Bousfield, A. K., Constructions of factorization systems in categories, J. Pure Appl. Algebra, 9, 2, 207-220 (197677) · Zbl 0361.18001 · doi:10.1016/0022-4049(77)90067-6 |
| [4] | Bucur, I.; Deleanu, A., Introduction to the theory of categories and functors, XIX, x+224 pp. (1968), Intersci. Publ. John Wiley & Sons, Ltd.: Intersci. Publ. John Wiley & Sons, Ltd., London-New York-Sydney · Zbl 0197.29205 |
| [5] | Deligne, P., Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math., 44, 5-77 (1974) · Zbl 0237.14003 · doi:10.1007/BF02685881 |
| [6] | Dold, A., Homology of symmetric products and other functors of complexes, Ann. of Math. (2), 68, 54-80 (1958) · Zbl 0082.37701 · doi:10.2307/1970043 |
| [7] | Dwyer, W. G.; Hirschhorn, P. S.; Kan, D. M.; ; Smith, J. H., Homotopy limit functors on model categories and homotopical categories, 113, viii+181 pp. (2004), Amer. Math. Soc.: Amer. Math. Soc., Providence, RI · Zbl 1072.18012 · doi:10.1090/surv/113 |
| [8] | Dwyer, W. G.; Spali\'nski, J., Homotopy theories and model categories, Handbook of algebraic topology, 73-126 (1995), North-Holland: North-Holland, Amsterdam · Zbl 0869.55018 · doi:10.1016/B978-044481779-2/50003-1 |
| [9] | Gelfand, S. I.; Manin, Yu. I.; Gelfand, S. I.; Manin, Yu. I., Methods of homological algebra. Vol. I. Methods of homological algebra, xviii+372 pp. (1996), Nauka: Nauka, Moscow: Springer-Verlag, Nauka: Nauka, Moscow: Nauka: Nauka, Moscow: Springer-Verlag, Nauka: Nauka, Moscow, Berlin · Zbl 0855.18001 · doi:10.1007/978-3-662-03220-6 |
| [10] | Grothendieck, A., Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2), 9, 119-221 (1957) · Zbl 0118.26104 · doi:10.2748/tmj/1178244839 |
| [11] | Grothendieck, A., Catégories fibrées et descente, Revêtements étales et groupe fondamental, Séminaire de géométrie algébrique du Bois Marie 1960-1961 (SGA 1), 224, 145-194 (1971), Springer-Verlag: Springer-Verlag, Berlin-New York |
| [12] | Hovey, M., Model categories, 63, xii+209 pp. (1999), Amer. Math. Soc.: Amer. Math. Soc., Providence, RI · Zbl 0909.55001 · doi:10.1090/surv/063 |
| [13] | Jibladze, M.; Pirashvili, T., Cohomology of algebraic theories, J. Algebra, 137, 2, 253-296 (1991) · Zbl 0724.18005 · doi:10.1016/0021-8693(91)90093-N |
| [14] | Johnstone, P. T., Topos theory, 10, xxiii+367 pp. (1977), Academic Press: Academic Press, London-New York · Zbl 0368.18001 |
| [15] | Kaledin, D., Trace theories and localization, Stacks and categories in geometry, topology, and algebra, 643, 227-262 (2015), Amer. Math. Soc.: Amer. Math. Soc., Providence, RI · Zbl 1346.18027 · doi:10.1090/conm/643/12900 |
| [16] | Kaledin, D., How to glue derived categories, Bull. Math. Sci., 8, 3, 477-602 (2018) · Zbl 1435.18017 · doi:10.1007/s13373-018-0119-z |
| [17] | Kaledin, D. B.; Kaledin, D. B., Adjunction in 2-categories. Adjunction in 2-categories, Uspekhi Mat. Nauk. Russian Math. Surveys, 75, 5, 883-927 (2020) · Zbl 1477.18046 · doi:10.1070/RM9963 |
| [18] | Kaledin, D.; Lowen, W., Cohomology of exact categories and (non-)additive sheaves, Adv. Math., 272, 652-698 (2015) · Zbl 1304.18024 · doi:10.1016/j.aim.2014.11.016 |
| [19] | Kashiwara, M.; Schapira, P., Categories and sheaves, 332, x+497 pp. (2006), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1118.18001 · doi:10.1007/3-540-27950-4 |
| [20] | Keller, B., On differential graded categories, International Congress of Mathematicians. Vol. II, 151-190 (2006), Eur. Math. Soc.: Eur. Math. Soc., Zürich · Zbl 1140.18008 |
| [21] | Kontsevich, M.; Soibelman, Y.; Kontsevich, M.; Soibelman, Y., Notes on \(A_\infty \)-algebras, \(A_\infty \)-categories and non-commutative geometry. Notes on \(A_\infty \)-algebras, \(A_\infty \)-categories and non-commutative geometry, Homological mirror symmetry, 757, 153-219 (2006), Springer: Springer, Berlin · Zbl 1202.81120 · doi:10.1007/978-3-540-68030-7_6 |
| [22] | Lowen, W.; Van den Bergh, M., Deformation theory of abelian categories, Trans. Amer. Math. Soc., 358, 12, 5441-5483 (2006) · Zbl 1113.13009 · doi:10.1090/S0002-9947-06-03871-2 |
| [23] | Quillen, D. G., Homotopical algebra, 43, iv+156 pp. (1967), Springer-Verlag: Springer-Verlag, Berlin-New York · Zbl 0168.20903 · doi:10.1007/BFb0097438 |
| [24] | Quillen, D., Higher algebraic K-theory. I, Algebraic K-theory. Vol. I, 341, Higher K-theories, 85-147 (1973), Springer: Springer, Berlin · Zbl 0292.18004 · doi:10.1007/BFb0067053 |
| [25] | Sch\"appi, D., Ind-abelian categories and quasi-coherent sheaves, Math. Proc. Cambridge Philos. Soc., 157, 3, 391-423 (2014) · Zbl 1305.18043 · doi:10.1017/S0305004114000401 |
| [26] | Tamarkin, D., What do dg-categories form?, Compos. Math., 143, 5, 1335-1358 (2007) · Zbl 1138.18004 · doi:10.1112/S0010437X07002771 |
| [27] | Verdier, J. L., Topologies et faisceaux, Théorie de topos et cohomologie étale des schémas. Vol. 1, Séminaire de géométrie algébrique du Bois-Marie 1963-1964 (SGA 4), 269, 219-263 (1972), Springer-Verlag: Springer-Verlag, Berlin-New York · Zbl 0256.18006 · doi:10.1007/BFb0081553 |
| [28] | Verdier, J.-L., Des catégories dérivées des catégories abéliennes, With a preface by L. Illusie, edited and with a note by G. Maltsiniotis, 239, xii+253 pp. (1996), Soc. Math. France: Soc. Math. France, Paris · Zbl 0882.18010 · doi:10.24033/bsmf.1583 |
| [29] | Weibel, Ch. A., An introduction to homological algebra, 38, xiv+450 pp. (1994), Cambridge Univ. Press: Cambridge Univ. Press, Cambridge · Zbl 0797.18001 · doi:10.1017/CBO9781139644136 |
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