## A-infinity-bimodules and Serre A-infinity-functors

• This dissertation is intended to transport the theory of Serre functors into the context of A-infinity-categories. We begin with an introduction to multicategories and closed multicategories, which form a framework in which the theory of A-infinity-categories is developed. We prove that (unital) A-infinity-categories constitute a closed symmetric multicategory. We define the notion of A-infinity-bimodule similarly to Tradler and show that it is equivalent to an A-infinity-functor of two arguments which takes values in the differential graded category of complexes of k-modules, where k is a commutative ground ring. Serre A-infinity-functors are defined via A-infinity-bimodules following ideas of Kontsevich and Soibelman. We prove that a unital closed under shifts A-infinity-category over a field admits a Serre A-infinity-functor if and only if its homotopy category admits an ordinary Serre functor. The proof uses categories and Serre functors enriched in the homotopy category of complexes of k-modules. Another important ingredient is an A-infinity-version of the Yoneda Lemma.
• A-unendlich-Bimoduln und Serresche A-unendlich-Funktoren

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Verfasserangaben: Oleksandr Manzyuk urn:nbn:de:hbz:386-kluedo-21410 Gert-Martin Greuel Dissertation Englisch 2007 2007 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 25.10.2007 06.11.2007 A-infinity-bimodule; A-infinity-category; A-infinity-functor; Serre functor; multicategory Homologische Algebra; Kategorientheorie Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik 18-XX CATEGORY THEORY; HOMOLOGICAL ALGEBRA (For commutative rings see 13Dxx, for associative rings 16Exx, for groups 20Jxx, for topological groups and related structures 57Txx; see also 55Nxx and 55Uxx for algebraic topologyg) / 18Dxx Categories with structure / 18D10 Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18-XX CATEGORY THEORY; HOMOLOGICAL ALGEBRA (For commutative rings see 13Dxx, for associative rings 16Exx, for groups 20Jxx, for topological groups and related structures 57Txx; see also 55Nxx and 55Uxx for algebraic topologyg) / 18Dxx Categories with structure / 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.) 18-XX CATEGORY THEORY; HOMOLOGICAL ALGEBRA (For commutative rings see 13Dxx, for associative rings 16Exx, for groups 20Jxx, for topological groups and related structures 57Txx; see also 55Nxx and 55Uxx for algebraic topologyg) / 18Dxx Categories with structure / 18D20 Enriched categories (over closed or monoidal categories) 18-XX CATEGORY THEORY; HOMOLOGICAL ALGEBRA (For commutative rings see 13Dxx, for associative rings 16Exx, for groups 20Jxx, for topological groups and related structures 57Txx; see also 55Nxx and 55Uxx for algebraic topologyg) / 18Gxx Homological algebra [See also 13Dxx, 16Exx, 20Jxx, 55Nxx, 55Uxx, 57Txx] / 18G55 Homotopical algebra Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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