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

Details A-infinity-bimodules and Serre A-infinity-functors A-infinity-bimodules and Serre A-infinity-functors

2007-01-05

arxiv.org/abs/math/0701165v2

Geometry and Dynamics of Groups and Spaces, Progress in Math., 265, Birkhauser, Basel (2007) 565-645

Mathematics

Category Theory

122 pages, Latex + Paul Taylor's diagrams.sty. This is the published version + complete proof of A-infinity Yoneda Lemma

Scientific paper

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

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