Equivalence between k-compact generation of stable oo-categories and k-well generation of their homotopy categories

Establish that a stable presentable oo-category E is k-compactly generated if and only if its homotopy category Ho(E) is k-well generated for the same regular cardinal k; specifically, prove the converse implication that Ho(E) being k-well generated implies that E is k-compactly generated (and, more generally, that Lemma A.8(b) holds for any presentable E).

Background

The authors prove one direction (Proposition A.10): if E is k-compactly generated, then Ho(E) is k-well generated. They point out the desirability of a full equivalence for the same k but could only establish one implication.

Such an equivalence would align cardinal-relative generation notions between stable oo-categories and their triangulated homotopy categories, streamlining the transfer of structural results across settings.

References

By definition though, 'presentable' means k-compactly generated for some regular k; so it would be nice to show that & is k-compactly generated iff Ho(E) is k-well generated (for the same k!). This boils down to showing that part (b) of the lemma holds in a general presentable &. Unfortunately, we can only prove one implication.

— Stratification in equivariant Kasparov theory  (2412.21109 - Dell'Ambrogio et al., 2024) in Remark A.9, Appendix A

For uncountable regular cardinals \kappa, there is also a notion of \kappa-compact object in a triangulated category (Definition 4.2.7). For a stable \infty-category \cC and an object X \in \cC, we do not know how X being \kappa-compact in h\cC in this sense relates to X being \kappa-compact in \cC.

— Cocompactness and Presentability  (2609.20533 - Geiß et al., 17 Sep 2026) in Remark 2.13, Section 2, subsection on Cosumpactness and measurable cardinals