Directed a ∞ -subcategories and Natural Transformations

Let B be an A∞-category with finitely many objects (X1, . . . ,Xm). We impose some technical restrictions (finite-dimensional hom spaces, strict unitality). Consider the directed subcategory A ⊂ B, which retains only morphisms from Xi to Xj for i < j, together with the identity endomorphisms. Following [10], we associate to the pair (A,B) a further object D, which is a filtered A∞-category with nonzero curvature. Even though the definition of D is really simple, the effect of the curvature term is not obvious at first sight. To approach this question, we look at a suitable category modt(D) of modules over D (the “t” stands for torsion, with respect to the formal parameter defining the filtration of D). This is an ordinary differential graded category, and its cohomological category H0(modt(D)) is triangulated. A natural point of comparison is the corresponding category modf(A) (“f” means finite-dimensional), since that has the same formal properties and comes with a canonical functor