The Yoneda Lemma for unital A ∞ - categories
نویسنده
چکیده
Let C be the differential graded category of differential graded k-modules. 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], Lefèvre-Hasegawa [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 2-category, whose objects are unital A∞-categories, 1-morphisms are unital A∞-functors and 2-morphisms are equivalence classes of natural A∞-transformations [Lyu03]. We continue to study this 2-category. The notations and conventions are explained in the first section. Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine; [email protected] Department of Algebra, Faculty of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 64 Volodymyrska st., Kyiv, 01033, Ukraine; [email protected]
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A∞-categories and the Yoneda lemma
Let C be the differential graded category of differential graded k-modules. We prove that the Yoneda A ∞ -functor Y : A → A ∞ (A,C) is a full embedding for an arbitrary unital A ∞ -category A. For a differential graded k-quiver B we define the free A ∞ -category FB generated by B. The main result is that the restriction A ∞ -functor A ∞ (FB,A) → A1(B,A) is an equivalence, where objects of the l...
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