Ordinary-sheaf unbounded acyclic-pullback axiom

Establish whether, for every fibration X \to Y in the category of outer-unbounded Reedy-fibrant strict simplicial derived-smooth models with weakly represented levels and geometrically smooth-surjective positive relative homotopy horns, an ordinary realization equivalence X\to Y remains an ordinary realization equivalence after pullback along every map Z\to Y; equivalently, determine whether the ordinary realization weak-equivalence class satisfies the acyclic-pullback axiom \mathbf{AP}_\infty.

Background

The paper constructs a full Brown category of fibrant objects for outer-unbounded derived-smooth represented models when weak equivalences are defined by hypercompleted realization. Enriched realization base change proves that hypercompleted realization preserves the relevant pullbacks, thereby establishing stability of acyclic-fibration pullbacks for the hypercomplete weak-equivalence class.

The corresponding ordinary-sheaf realization class is not treated as established in the unbounded case. The unresolved issue is precisely whether an ordinary realization equivalence that is also a fibration remains an ordinary realization equivalence after arbitrary pullback. Resolving this would establish the remaining acyclic-pullback condition \mathbf{AP}_\infty and could yield the ordinary-sheaf unbounded Brown structure.

References

If a fibration $X\to Y$ in $\mathscr R_\infty$ is an ordinary realization equivalence, is $X\times_Y Z\to Z$ an ordinary realization equivalence for every $Z\to Y$? This is the precise remaining acyclic-pullback condition $\mathbf{AP}\infty$ for $(\mathscr R\infty,W_{\mathrm{sh}},F)$.

— Derived Smooth and Banach Higher Groupoids: Representability and Descent  (2609.28220 - Zeng, 23 Sep 2026) in Section 3, subsection “Finite ordinary realization and unbounded hypercomplete realization,” Question 3.??, labeled “The ordinary-sheaf unbounded axiom” (immediately after Theorem “Unbounded hypercomplete represented CFO”)