On the tensor product of well generated dg categories. (English) Zbl 1472.18010
Summary: We endow the homotopy category of well generated (pretriangulated) dg categories with a tensor product satisfying a universal property. The resulting monoidal structure is symmetric and closed with respect to the cocontinuous RHom of dg categories (in the sense of B. Toën [Invent. Math. 167, No. 3, 615–667 (2007; Zbl 1118.18010)]). We give a construction of the tensor product in terms of localisations of dg derived categories, making use of the enhanced derived Gabriel-Popescu theorem [M. Porta, Adv. Math. 225, No. 3, 1669–1715 (2010; Zbl 1227.18011)]. Given a regular cardinal \(\alpha\), we define and construct a tensor product of homotopically \(\alpha\)-cocomplete dg categories and prove that the well generated tensor product of \(\alpha\)-continuous derived dg categories (in the sense of Porta [loc. cit.]) is the \(\alpha\)-continuous dg derived category of the homotopically \(\alpha\)-cocomplete tensor product. In particular, this shows that the tensor product of well generated dg categories preserves \(\alpha\)-compactness.
MSC:
| 18G80 | Derived categories, triangulated categories |
| 18G35 | Chain complexes (category-theoretic aspects), dg categories |
| 18M05 | Monoidal categories, symmetric monoidal categories |
| 18E35 | Localization of categories, calculus of fractions |
References:
| [1] | Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, (Artin, M.; Grothendieck, A.; de Verdier, J. L., Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4). Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Lect. Notes Math., vol. 269 (1972), Springer-Verlag: Springer-Verlag Berlin-New York), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652 · Zbl 0234.00007 |
| [2] | Adámek, J.; Rosický, J., Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, vol. 189 (1994), Cambridge University Press: Cambridge University Press Cambridge, MR 1294136 · Zbl 0795.18007 |
| [3] | Ben-Bassat, O.; Block, J., Milnor descent for cohesive dg-categories, J. K-Theory, 12, 3, 433-459 (2013), MR 3165183 · Zbl 1296.18005 |
| [4] | Bondal, A. I., Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR, Ser. Mat., 53, 1, 25-44 (1989), MR 992977 |
| [5] | Bondal, A. I.; Kapranov, M. M., Representable functors, Serre functors, and mutations, Math. USSR, Izv., 35, 3, 519-541 (1990) · Zbl 0703.14011 |
| [6] | Canonaco, A.; Stellari, P., Internal Homs via extensions of dg functors, Adv. Math., 277, 100-123 (2015), MR 3336084 · Zbl 1355.14012 |
| [7] | Canonaco, A.; Stellari, P., Uniqueness of dg enhancements for the derived category of a Grothendieck category, J. Eur. Math. Soc., 20, 11, 2607-2641 (2018), MR 3861804 · Zbl 1427.14039 |
| [8] | Cohn, L., Differential graded categories are k-linear stable infinity categories (2016) |
| [9] | Drinfeld, V., DG quotients of DG categories, J. Algebra, 272, 2, 643-691 (2004), MR 2028075 · Zbl 1064.18009 |
| [10] | Genovese, F., Adjunctions of quasi-functors between DG-categories, Appl. Categ. Struct., 25, 4, 625-657 (2017), MR 3669175 · Zbl 1387.18006 |
| [11] | Iyengar, S. B.; Krause, H., The Bousfield lattice of a triangulated category and stratification, Math. Z., 273, 3-4, 1215-1241 (2013), MR 3030697 · Zbl 1328.18017 |
| [12] | Keller, B., Deriving DG categories, Ann. Sci. Éc. Norm. Supér. (4), 27, 1, 63-102 (1994), MR 1258406 · Zbl 0799.18007 |
| [13] | Keller, B., On the cyclic homology of exact categories, J. Pure Appl. Algebra, 136, 1, 1-56 (1999), MR 1667558 · Zbl 0923.19004 |
| [14] | Keller, B., On differential graded categories, (International Congress of Mathematicians. Vol. II (2006), Eur. Math. Soc.: Eur. Math. Soc. Zürich), 151-190, MR 2275593 · Zbl 1140.18008 |
| [15] | Kelly, G. M., Structures defined by finite limits in the enriched context. I, Third Colloquium on Categories, Part VI (Amiens, 1980). Third Colloquium on Categories, Part VI (Amiens, 1980), Cah. Topol. Géom. Différ. Catég., 23, 1, 3-42 (1982), MR 648793 · Zbl 0538.18006 |
| [16] | Kelly, G. M., Basic concepts of enriched category theory, Repr. Theory Appl. Categ., 10 (2005), vi+137, Reprint of the 1982 original [Cambridge Univ. Press, Cambridge; MR0651714] · Zbl 1086.18001 |
| [17] | Krause, H., Localization Theory for Triangulated Categories, Triangulated Categories, London Math. Soc. Lecture Note Ser., vol. 375, 161-235 (2010), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, MR 2681709 · Zbl 1232.18012 |
| [18] | Kuznetsov, A., Base change for semiorthogonal decompositions, Compos. Math., 147, 3, 852-876 (2011), MR 2801403 · Zbl 1218.18009 |
| [19] | López Franco, I., Tensor products of finitely cocomplete and abelian categories, J. Algebra, 396, 207-219 (2013), MR 3108080 · Zbl 1305.18042 |
| [20] | Lowen, W.; Ramos González, J.; Shoikhet, B., On the tensor product of linear sites and Grothendieck categories, Int. Math. Res. Not., 2018, 21, 6698-6736 (2018), MR 3873542 · Zbl 1408.18024 |
| [21] | Lunts, V. A.; Orlov, D. O., Uniqueness of enhancement for triangulated categories, J. Am. Math. Soc., 23, 3, 853-908 (2010), MR 2629991 · Zbl 1197.14014 |
| [22] | Lurie, J., Derived algebraic geometry II: noncommutative algebra (2007) |
| [23] | Lurie, J., Higher algebra (2017) |
| [24] | Neeman, A., On the derived category of sheaves on a manifold, Doc. Math., 6, 483-488 (2001), MR 1874232 · Zbl 1026.18006 |
| [25] | Neeman, A., Triangulated Categories, Annals of Mathematics Studies, vol. 148 (2001), Princeton University Press: Princeton University Press Princeton, NJ, MR 1812507 · Zbl 0974.18008 |
| [26] | Popesco, N.; Gabriel, P., Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes, C. R. Math. Acad. Sci. Paris, 258, 4188-4190 (1964), MR 166241 · Zbl 0126.03304 |
| [27] | Porta, M., The Popescu-Gabriel theorem for triangulated categories, Adv. Math., 225, 3, 1669-1715 (2010), MR 2673743 · Zbl 1227.18011 |
| [28] | Positselski, L.; Rosický, J., Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories, J. Algebra, 483, 83-128 (2017), MR 3649814 · Zbl 1402.18011 |
| [29] | Robalo, M., Motivic homotopy theory of noncommutative spaces (2014), Université Montpellier 2, Ph.D. thesis |
| [30] | Tabuada, G., Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris, 340, 1, 15-19 (2005), MR 2112034 · Zbl 1060.18010 |
| [31] | Tabuada, G., On Drinfeld’s dg quotient, J. Algebra, 323, 5, 1226-1240 (2010), MR 2584954 · Zbl 1244.14002 |
| [32] | Toën, B., The homotopy theory of dg-categories and derived Morita theory, Invent. Math., 167, 3, 615-667 (2007), MR 2276263 · Zbl 1118.18010 |
| [33] | Toën, B., Derived Azumaya algebras and generators for twisted derived categories, Invent. Math., 189, 3, 581-652 (2012), MR 2957304 · Zbl 1275.14017 |
| [34] | Williams, N. H., On Grothendieck universes, Compos. Math., 21, 1-3 (1969), MR 0244035 · Zbl 0175.00701 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
