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DG quotients of DG categories
 J. Algebra
"... Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. ..."
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Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.
Notes on A∞algebras, A∞categories and noncommutative geometry, Homological mirror symmetry
 Lecture Notes in Phys
, 2009
"... 1.1 A∞algebras as spaces........................ 2 ..."
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1.1 A∞algebras as spaces........................ 2
Contents QUOTIENTS OF UNITAL A∞CATEGORIES
"... Abstract. Assuming that B is a full A∞subcategory of a unital A∞category C we construct the quotient unital A∞category D =‘C/B’. It represents the Au ∞2functor A ↦ → Au ∞(C, A)mod B, which associates with a given unital A∞category A the A∞category ..."
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Abstract. Assuming that B is a full A∞subcategory of a unital A∞category C we construct the quotient unital A∞category D =‘C/B’. It represents the Au ∞2functor A ↦ → Au ∞(C, A)mod B, which associates with a given unital A∞category A the A∞category
Free A∞categories
, 2004
"... For a differential graded kquiver Q we define the free A∞category FQ generated by Q. The main result is that the restriction A∞functor A∞(FQ,A) → A1(Q,A) is an equivalence, where objects of the last A∞category are morphisms of differential graded kquivers Q → A. A∞categories defined by Fukaya ..."
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For a differential graded kquiver Q we define the free A∞category FQ generated by Q. The main result is that the restriction A∞functor A∞(FQ,A) → A1(Q,A) is an equivalence, where objects of the last A∞category are morphisms of differential graded kquivers Q → A. A∞categories defined by Fukaya [Fuk93] and Kontsevich [Kon95] are generalizations of differential graded categories for which the binary composition is associative only up to a homotopy. They also generalize A∞algebras introduced by Stasheff [Sta63, II]. A∞functors are the corresponding generalizations of usual functors, see e.g. [Fuk93, Kel01]. Homomorphisms of A∞algebras (e.g. [Kad82]) are particular cases of A∞functors. A∞transformations are certain coderivations. Examples of such structures are encountered in studies of mirror symmetry (e.g. [Kon95, Fuk02]) and in homological algebra. For an A∞category there is a notion of units up to a homotopy (homotopy identity morphisms) [Lyu03]. Given two A∞categories A and B, one can construct a third A∞category A∞(A, B), whose objects are A∞functors f: A → B, and morphisms are A∞transformations between such functors (Fukaya [Fuk02], Kontsevich and Soibelman [KS02, KS], LefèvreHasegawa [LH03], as well as [Lyu03]). This allows to define a 2category, whose objects are unital A∞categories, 1morphisms are unital A∞functors and 2morphisms are equivalence classes of natural A∞transformations [Lyu03]. We continue to study this 2category. The notations and conventions are explained in the first section. We also describe ANcategories, ANfunctors and ANtransformations – truncated at N < ∞ versions of A∞categories. For instance, A1categories and A1functors are differential graded kquivers and their morphisms. However, A1transformations bring new 2categorical features to the theory. In particular, for any differential graded kquiver Q and any A∞category
The Yoneda Lemma for unital A∞categories
, 2003
"... Let C be the differential graded category of differential graded kmodules. We prove that the Yoneda A∞functor Y: A op → A∞(A,C) is a full embedding for an arbitrary unital A∞category A. Since A∞algebras were introduced by Stasheff [Sta63, II] there existed a possibility to consider A∞generaliza ..."
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Let C be the differential graded category of differential graded kmodules. We prove that the Yoneda A∞functor Y: A op → A∞(A,C) is a full embedding for an arbitrary unital A∞category A. Since A∞algebras were introduced by Stasheff [Sta63, II] there existed a possibility to consider A∞generalizations of categories. It did not happen until A∞categories were encountered in studies of mirror symmetry by Fukaya [Fuk93] and Kontsevich [Kon95]. A∞categories may be viewed as generalizations of differential graded categories for which the binary composition is associative only up to a homotopy. The possibility to define A∞functors was mentioned by Smirnov [Smi89], who reformulated one of his results in the language of A∞functors between differential graded categories. The definition of A∞functors between A∞categories was published by Keller [Kel01], who studied their applications to homological algebra. Homomorphisms of A∞algebras (e.g. [Kad82]) are particular cases of A∞functors. A∞transformations between A∞functors are certain coderivations. Given two A∞categories A and B, one can construct a third A∞category A∞(A, B), whose objects are A∞functors f: A → B, and morphisms are A∞transformations (Fukaya [Fuk02], Kontsevich and Soibelman [KS02, KS], LefèvreHasegawa [LH03], as well as [Lyu03]). For an A∞category there is a notion of units up to a homotopy (homotopy identity morphisms) [Lyu03]. This allows to define a 2category, whose objects are unital A∞categories, 1morphisms are unital A∞functors and 2morphisms are equivalence classes of natural A∞transformations [Lyu03]. We continue to study this 2category. The notations and conventions are explained in the first section.
VOLODYMYR LYUBASHENKO AND OLEKSANDR MANZYUK
, 802
"... Abstract. We prove that three definitions of unitality for A∞categories suggested by the first author, by Kontsevich and Soibelman, and by Fukaya are equivalent. ..."
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Abstract. We prove that three definitions of unitality for A∞categories suggested by the first author, by Kontsevich and Soibelman, and by Fukaya are equivalent.