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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
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
QUOTIENTS OF UNITAL A∞CATEGORIES
, 2008
"... 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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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
UNIVERSITÉ DE NICESOPHIA ANTIPOLIS — UFR Sciences École Doctorale de Sciences Fondamentales et Appliquées THÈSE pour obtenir le titre de Docteur en Sciences
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The homotopy theory of dgcategories and derived
, 2006
"... Abstract. The main purpose of this work is to study the homotopy theory of dgcategories up to quasiequivalences. Our main result is a description of the mapping spaces between two dgcategories C and D in terms of the nerve of a certain category of (C, D)bimodules. We also prove that the homotopy ..."
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Abstract. The main purpose of this work is to study the homotopy theory of dgcategories up to quasiequivalences. Our main result is a description of the mapping spaces between two dgcategories C and D in terms of the nerve of a certain category of (C, D)bimodules. We also prove that the homotopy category Ho(dg − Cat) possesses internal Hom’s relative to the (derived) tensor product of dgcategories. We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dgcategories of modules over two dgcategories C and D as the dgcategory of (C, D)bimodules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the classifying space of dgcategories (i.e. the nerve of the category of dgcategories and quasiequivalences between them). The second application is the existence of a good theory of localization for dgcategories, defined in terms of a natural universal property. Our last application states that the dgcategory of
Unital A∞categories
, 2008
"... 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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We prove that three definitions of unitality for A∞categories suggested by the first author, by Kontsevich and Soibelman, and by Fukaya are equivalent.
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.