On the A-infinity Representations of Semifree DG Categories
In general, an A-infinity natural transformation between dg functors consists of infinitely many morphisms. We show that if the domain of the dg functors is a “semifree” dg category C, then an A-infinity natural transformation can be simply described by a morphism for each object and for each generating morphism of C. This is closely related to the result by Karabas-Lee. This result has some applications to symplectic geometry as follows:
Fukaya categories of a Weinstein manifold W embed inside the dg category of A-infinity modules over the dg category C via Yoneda embedding, where the objects of C are the Lagrangian cocores of W, and the morphisms are wrapped Floer chain complexes between them. This means that a Floer chain complex of two Lagrangians can be expressed by a chain complex created from A-infinity natural transformations between their representations (i.e. corresponding A-infinity modules). If C is semifree, A-infinity modules can be seen as dg functors with the domain C, and then by our result, one can study the Floer homology of Lagrangians easily via simplified A-infinity transformations between their representations.
Note that the Floer homology of an embedded Lagrangian with itself gives the cohomology of the Lagrangian. If the Lagrangian is immersed, Floer homology and singular cohomology are still related. Then, as an application, we can study the representations of all Lagrangians to show that Lagrangians of a particular homotopy type cannot exist in a given symplectic manifold.
Note also that Hochschild cohomology of an A-infinity category C can be described as the cohomology of the chain complex of A-infinity natural transformations from the identity functor to itself. Then, as the second application, one can use our description to give an alternative formula for Hochschild cohomology when C is a semifree dg category. When C consists of Lagrangian cocores in W as before, it is pretriangulated equivalent to the wrapped Fukaya category of W, hence the Hochschild cohomology of C gives the symplectic cohomology of W.
Draft of the current results is available upon request.