Morita invariance of Aut^{∞}_{∘}(C)

Establish that the subgroup Aut^{∞}_{∘}(C) of weak equivalence classes of A∞-endofunctors of a graded k-linear category C (with finite-dimensional category algebra) whose first Taylor coefficient lies in the identity component of the outer automorphism group Out^{∘}(C) is a Morita invariant; that is, prove that if graded k-linear categories C and D are Morita equivalent, then Aut^{∞}_{∘}(C) ≅ Aut^{∞}_{∘}(D).

Background

The paper introduces Aut^{∞}_{∘}(C) as a natural candidate for the identity component of the derived Picard group DPic(C), built from A∞-endofunctors whose linear part is in the identity component of Out(C). This group fits into a split exact sequence with Aut^{∞}_+(C) (A∞-isotopies) and Out^{∘}(C), and is amenable to computation via Hochschild cohomology in characteristic zero.

The author notes that, while this paper verifies Morita invariance of Aut^{∞}_{∘}(C) for graded gentle algebras, the general case remains unresolved. A conceptual obstacle is that showing Morita invariance implicitly requires that every element in the identity component of DPic(C) be representable by an A∞-endofunctor of C, which may fail in general, as illustrated by examples.