Necessity of large presentable categories

Determine whether the use of large stable presentable infinity-categories is necessary for the proof and formulation of the perverse-schober construction, particularly for the application of the Kleisli–Eilenberg–Moore equivalence in Proposition 3.1.

Background

The paper works with large stable presentable infinity-categories because Proposition 3.1 identifies the Kleisli and Eilenberg–Moore constructions in that setting. This largeness assumption is used in the analysis of perverse schobers supported on the cuspidal cubic y2=x3.

The authors explicitly leave unresolved whether the same argument can be carried out in a smaller categorical framework, so the necessity of the largeness hypothesis is an open technical question.

References

We show that our definition of $2Perv(C2_{x,y},C_{2,3})$ decategorifies to Proposition~\ref{prop:mv}. Note however that we must be in the setting of large categories in order to use Proposition~\ref{prop:KLisEM}, although it is not clear if this is necessary.

— Microlocal perverse schobers and Radon transform  (2609.19692 - Okitani, 17 Sep 2026) in Remark following Proposition 5.8, Section 5, subsection “Schobers supported on a cuspidal cubic”