Compact generation or compact assembly of KK from [KKG]

Ascertain whether the stable ∞-category KK (as constructed in [KKG]) is compactly generated or at least compactly assembled; explicitly, either prove compact generation or, failing that, establish compact assembly for KK.

Background

The paper develops a framework for compactly assembled ∞-categories and notes that certain results would simplify if one could assume such a structure for KK from [KKG]. The authors point out elsewhere that KKG_sep is not currently known to be compactly assembled.

Establishing compact generation or compact assembly for KK would enable broader application of the techniques introduced here and could clarify the role of KK within the broader landscape of triangulated and stable ∞-categories in noncommutative geometry.

References

At the end of this introduction we list some open questions.

  1. Is the stable $\infty$-category $KK$ from compactly generated or at least compactly assembled?
— $E$-theory is compactly assembled (2402.18228 - Bunke et al., 28 Feb 2024) in Introduction (end)