A∞-bimodules and Serre A∞-functors

  • Volodymyr Lyubashenko
  • Oleksandr Manzyuk
Chapter
Part of the Progress in Mathematics book series (PM, volume 265)

Abstract

We define A ∞-bimodules similarly to Tradler and show that this notion is equivalent to an A ∞-functor with two arguments which takes values in the differential graded category of complexes of \( \Bbbk \)-modules, where \( \Bbbk \) is a ground commutative ring. Serre A ∞-functors are defined via A ∞-bimodules likewise Kontsevich and Soibelman. We prove that a unital closed under shifts A ∞-category \( \mathcal{A} \) over a field \( \mathcal{A} \) admits a Serre A ∞-functor if and only if its homotopy category H 0 \( \mathcal{A} \) admits a Serre \( \Bbbk \)-linear functor. The proof uses categories enriched in \( \mathcal{K} \), the homotopy category of complexes of \( \Bbbk \)-modules, and Serre \( \mathcal{K} \)-functors. Also we use a new A ∞-version of the Yoneda Lemma generalizing the previously obtained result.

Keywords

A∞-categories A∞-modules A∞-bimodules Serre A∞-functors Yoneda Lemma 
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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Volodymyr Lyubashenko
    • 1
  • Oleksandr Manzyuk
    • 2
  1. 1.Institute of Mathematics NASUKyiv-4Ukraine
  2. 2.Fachbereich MathematikKaiserslauternGermany

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