Equivalence criterion for mutually connected and truncated morphisms of infinity-groupoids

Determine whether, for tame infinity-groupoids with finitely many connected components, the existence of morphisms in both directions that are respectively $(n-1)$-connected and $n$-truncated, together with the resulting isomorphism of all homotopy groups, implies that the morphisms are equivalences.

Background

The paper extends properties of groupoid cardinality and factorization systems from (2,1)-categories to tame infinity-groupoids. It observes that if tame infinity-groupoids with finitely many connected components admit morphisms in both directions that are (n−1)(n-1)-connected and nn-truncated, then the two objects have isomorphic homotopy groups in every degree. The unresolved issue is whether this homotopy-group-level agreement is sufficient to conclude that the given morphisms are equivalences, providing an infinity-categorical analogue of the equivalence criterion established earlier for tame groupoids.

References

Regarding Theorem \ref{equi_if_tame}, it is clear that if $X$ and $Y$ are tame $\infty$-groupids with only finitely many connected components and we have morphisms $X \to Y$ and $Y \to X$ which are $(n-1)$-connected and $n$-truncated, then $X$ and $Y$ have all isomorphic homotopy groups. However, it remains open whether this would imply that the morphisms in question are equivalences.

— Combinatorics in (2,1)-categories  (2502.03585 - Zehr, 5 Feb 2025) in Section 6, “Postnikov systems and homotopy cardinality”