The local splitting statement over the ind-completion #
The last hop of Deligne's 2.10: over the ind-completion of a small rigid abelian tensor category, the unit of the splitting algebra survives — the colimit unit dies only at a finite stage by the filtered criterion, and no stage unit of a monic point vanishes. The local splitting statement therefore holds for every short exact sequence whose relevant objects carry duals.
The unit of the splitting algebra survives over the ind-completion: the colimit unit dies only at a finite stage, and the stage units are nonvanishing symmetrised point powers of the monic dualised point.
The local splitting statement over the ind-completion (Deligne 2.10): every short exact sequence whose quotient and derived objects carry duals splits after base change to a nonzero commutative algebra.