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t-exactness of the universal homotopy ind-dg-completion

Determine whether the dg-functor Ind^{dg,Q}(F): Ind^{dg,Q}(A) → Ind^{dg,Q}(B) provided by the universal property of the homotopy ind-dg-completion is t-exact whenever F: A → B is a t-exact dg-functor between essentially small strongly pretriangulated t-dg-categories.

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Background

The universal property (Corollary 1.3, cor - universal prop inddgho) constructs a dg-functor Ind{dg,Q}(F) extending F to the homotopy ind-dg-completions, which are dg-enhancements of the derived categories built from filtered homotopy dg-colimits of representables.

While Corollary 1.4 (cor - D(F) t-exact) shows that a related extension D(F) is t-exact, the authors note that it is unclear whether the universally defined Ind{dg,Q}(F) itself preserves t-structures. Establishing t-exactness (or finding necessary/sufficient criteria) would strengthen the compatibility of the homotopy ind-dg-completion with t-structures.

References

We do not seem to have t-exactness in the universal property of \Cref{cor - universal prop inddgho} in general: even if $F$ in eq - universal prop enriched is $t$-exact, it is unclear whether $\Ind{\dg,Q}(F)$ is, because taking the homotopy ind-dg-completion strictly enlarges the category.

— Deformations of triangulated categories with t-structures via derived injectives (2411.15359 - Genovese et al., 22 Nov 2024) in Section “An induced t-structure on the derived category,” just before Corollary “D(F) t-exact”