A∞-subalgebras and Natural Transformations

The setup is as follows. We consider pairs (A,B) consisting of an A∞-algebra B and a subalgebra A. We will be primarily working with the category V = H0(mod(A)) of Amodules. However, that category inherits certain additional data from the presence of B. The most important such data are a functor F : V → V , given by tensor product with the quotient bimodule (B/A)[−1], together with a natural transformation T : F → Id.