Quasi-equivalence of the inclusion I_{(TwA)*G} into Tw((TwA)*G)

Determine whether the natural inclusion functor I_{(TwA)*G}:(TwA)*G→Tw((TwA)*G) is a quasi-equivalence for an arbitrary strictly unital A_infinity-category A equipped with a strict action of a finite group G whose order is not divisible by the characteristic of the base field K. Equivalently, ascertain whether the skew-group A_infinity-category (TwA)*G is triangulated so that its canonical embedding into its twisted-complex category is a quasi-equivalence.

Background

In Proposition \ref{prop::comm-square}, the authors construct a commutative square of strict A_infinity-functors involving A*G, (TwA)*G, Tw(A*G), and Tw((TwA)*G), and establish that the bottom arrow is a quasi-equivalence and that the vertical arrows are Morita equivalences. The right-hand arrow in that square is the canonical embedding I_{(TwA)*G}:(TwA)*G→Tw((TwA)*G).

For a strictly unital A_infinity-category B, it is known (Seidel) that B is triangulated if and only if the embedding I_B:B→TwB is a quasi-equivalence. Thus, asking whether I_{(TwA)*G} is a quasi-equivalence amounts to asking whether (TwA)*G is triangulated. In the subsequent section, the authors show that after passing to split-closures, the corresponding right-most arrow becomes a quasi-equivalence, but they leave open the question in the original (non-split-closed) setting.

References

In \cref{prop::comm-square}, it is unclear whether the right arrow should be a quasi-equivalence. In the next section, we will see that it will become one if we pass to split-closures.

— Skew-group $A_{\infty}$-categories as Fukaya categories of orbifolds  (2405.15466 - Amiot et al., 2024) in Remark following Proposition 2.?, Section 2 (Skew-group A_infinity-categories)

Sun proved uniqueness once the canonical triangulation exists and left its existence open in general Corollary~3.6 and the paragraph following Example~3.8.

— From Pre-triangulation to Triangulation: Obstructions and Exact Lifting  (2609.08684 - He et al., 8 Sep 2026) in Section 9, subsection “Equivariantization”