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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Author:Oleksandr Manzyuk
URN (permanent link):urn:nbn:de:hbz:386-kluedo-21410
Advisor:Gert-Martin Greuel
Document Type:Doctoral Thesis
Language of publication:English
Year of Completion:2007
Year of Publication:2007
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2007/10/25
Tag:A-infinity-bimodule; A-infinity-category; A-infinity-functor; Serre functor; multicategory
GND-Keyword:Homologische Algebra; Kategorientheorie
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:510 Mathematik
MSC-Classification (mathematics):18D10 Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
18D20 Enriched categories (over closed or monoidal categories)
18G55 Homotopical algebra

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