1. Introduction1.1. 1.2. 1.3. Hom complexes of the DG quotient1.4. Proposition1.5. On Keller's construction of the DG quotient1.6. Universal property of the DG quotient1.7. More on uniqueness1.8. What do DG categories form?1.9. Structure of the article1.10. 2. DG categories: recollections and notation2.1. 2.2. Remark2.3. 2.4. 2.5. Proposition2.6. Remark2.7. Derived category of DG modules2.8. 2.9. 2.10. 3. A new construction of the DG quotient3.1. Construction3.2. Example3.3. 3.4. Theorem3.5. 3.6. Remarks3.7. Example4. The DG categories A and A. Keller's construction of the DG quotient.4.1. 4.2. Remark4.3. Remark4.4. Lemma4.5. 4.6. Proposition4.7. Proposition4.8. Remark4.9. 4.10. 4.11. Remarks4.12. 4.13. 5. Derived DG functors5.1. Deligne's definition5.2. 5.3. 5.4. Proposition.5.5. 5.6. 6. Some commutative diagrams6.1. Uniqueness of DG quotient6.2. More diagrams (to be used in §7)7. More on derived DG functors.7.1. 7.2. 7.3. 7.4. 7.5. 8. Proof of Theorem 3.4.8.1. 8.2. 8.3. 8.4. Lemma.8.5. Lemma8.6. Proof of Lemma 8.49. Proof of Propositions 1.4 and 5.4.9.1. Proof of Proposition 5.49.2. Lemma9.3. Proof of Proposition 1.410. Proof of Propositions 1.5.1, 4.6 and 4.7.10.1. Proof of Proposition 4.710.2. Proof of Proposition 1.5.110.3. Proof of Proposition 4.611. Proof of Proposition 1.6.3 and Theorem 1.6.211.1. Proof of Proposition 1.6.311.2. Proof of Theorem 1.6.212. Appendix I: Triangulated categories.12.1. Categories with Z-action and graded categories12.2. Quotients12.3. Remarks12.4. 12.5. Lemma12.6. Admissible subcategories13. Appendix II: Semi-free resolutions.13.1. Definition13.2. Remarks13.3. Lemma13.4. 13.5. Lemma13.6. Lemma14. Appendix III: DG modules over DG categories14.1. 14.2. 14.3. 14.4. Example14.5. 14.6. 14.7. 14.8. 14.9. 14.10. Example14.11. Example14.12. Derived induction14.13. 14.14. 14.15. Lemma14.16. Quasi-representability15. Appendix IV: The diagonal DG categories15.1. 15.2. 15.3. Lemma16. Appendix V: The 2-category of DG categories16.1. Flat case16.2. Remark16.3. General case16.4. 16.5. Remarks16.6. Ind-version and duality16.7. Relation with Kontsevich's approach16.8. DG models of T(A1,A2)16.9. Some historical remarksReferencesarXiv:math.KT/0210114 v6 20 May 2003DG QUOTIENTS OF DG CATEGORIESVLADIMIR DRINFELDAbstract. Keller introduced a notion of quotient of a differential gradedcategory modulo a full differential graded subcategory which agrees withVerdier’s notion of qu otient of a triangulated category modulo a trian-gulated subcategory. This work is an attempt to further develop histheory.Key words: DG category, triangulated category, derived category,localizationConventions. We fix a commutative ring k and write ⊗ instead of ⊗kand“DG category” instead of “ differential graded k-category”. If A is a DGcategory we write “DG mo dule over A” instead of “DG f unctor from A to theDG category of complexes of k-modules” (more details on the DG moduleterminology can be found in §14). Unless stated otherwise, all categories areassumed to be small. Triangulated categories are s ystematically viewed as Z-graded categories (see 12.1). A triangulated subcategory C′of a triangulatedsubcategory C is required to be full, but we do not require it to be strictlyfull (i.e., to contain all objects of C isomorphic to an object of C′). Inthe definition of quotient of a triangulated category we do not require thesubcategory to be thick (see 12.2-12.3).1. Introduction1.1. It has been clear to the experts since the 1960’s that Verdier’s notionsof derived category and triangulated category [56, 57] are not quite satisfac-tory: when you pass to the homotopy category you forget too much. Thisis why Grothendieck developed his derivator theory [17, 40].A different approach was suggested by Bondal and Kapranov [4]. Ac-cording to [4] one should work with pretriangulated DG categories ratherthan with triangulated categories in Verdier’s sense (e.g., with the DG cat-egory of bounded above complexes of projective modules rather than thebounded above derived category of modules). Hopefully the part of homo-logical algebra most relevant for algebraic geometry will be rewritten usingDG categories or r ather the more flexible notion of A∞-category due toFukaya and Kontsevich (see [14, 15, 30, 31, 24, 25, 33, 36, 37]), which goesback to Stasheff’s notion of A∞-algebra [51, 52].One of the basic tools developed by Verdier [56, 57] is the notion of quo-tient of a triangulated category by a triangulated subcategory. Keller [23]Partially supported by NSF grant DMS-0100108.12 VLADIMIR DRINFELDhas started to develop a theory of quotients in the DG setting. This workis an attempt to further develop his theory. I tried to make this articleessentially self-contained, in particular it can be r ead independently of [23].The notion of quotient in the setting of A∞-categories is being developedby Kontsevich – Soibelman [33] and Lyubashenko – Ovsienko [38]).1.2. The basic notions related to that of DG category are recalled in §2.Let A be a DG category and B ⊂ A a full DG subcategory. Let Atrdenotethe triangulated category associated to A (we recall its definition in 2.4).A DG quotient (or simply a q uotient ) of A modulo B is a diagram of DGcategories and DG functors(1.1) A≈←−˜Aξ→ Csuch that the DG functor˜A → A is a quasi-equivalence (see 2.3 for thedefinition), the functor Ho(˜A) → Ho(C) is essentially surjective, and thefunctor˜Atr→ Ctrinduces an equivalence Atr/Btr→ Ctr. Keller [23] provedthat a DG quotient always exists (recall that our DG categories are assumedto be small, otherwise even the existence of Atr/Btris not clear). We recallhis construction of the DG quotient in §4, and give a new construction in§3.The new construction is reminiscent of but easier than Dwyer-Kan local-ization [11, 12, 13]. It is very simple under a certain flatness assumption(which is satisfied automatically if one works over a field): one just kills theobjects of B (see 3.1). Without this assumption one h as to fi rst replace Aby a suitable resolution (see 3.5).The id ea of Keller’s original construction of the DG quotient (see §4)is to take the orthogonal complement of B as a DG quotient, but as theorthogonal complement of B in A is not necessarily big enough he takesthe complement not in A but in its ind -versionA→studied by him in [22].The reason why it is natural to consider the orthogonal complement inA→isexplained in 1.5. Of course,