Projectives and enough projectives in LB-based exact categories

Characterize all projective objects and ascertain whether there are enough projectives in the categories (LB,C) with C ∈ {D, E, E} and in (LB,E) with E ∈ {E, E}.

Background

The authors completely classify projectives in (LB,D) and show the existence of enough projectives there, but for other exact/deflation-exact structures on LB and its subcategories, projective classification and existence of enough projectives remain unsettled. Resolving these questions is important for computing global dimensions and understanding derived equivalences via projective resolutions.

References

In the categories $(LB,C)$ with $C\in{D,E,E}$ and $(LB,E)$ with $E\in{E,E}$ any object that is isomorphic to a countable direct sum of $\ell1(\Gamma)$s is projective. It is unknown if these are all projectives and if these categories have enough projectives.

— A homological approach to (Grothendieck's) completeness problem for regular LB-spaces  (2512.13161 - Wegner, 15 Dec 2025) in Remark after Theorem PROP-PROJ-2, Section 9 (Global dimensions)