Equivalence of Haugseng, Gwilliam–Scheimbauer, and bi-bimodule higher Morita models

Prove that, for every n and every k≤n, the k-morphism categories of Haugseng’s higher Morita model, the Gwilliam–Scheimbauer higher Morita model, and the bi-bimodule model are equivalent: hom^k_{Mrt^{Haug}_{E_n}(C)} ≃ hom^k_{Mrt^{GS}_{E_n}(C)} ≃ hom^k_{Mrt_{E_n}(C)}.

Background

The paper proves levelwise equivalences between the Haugseng and Gwilliam–Scheimbauer constructions and, for n=2, also identifies the resulting morphism categories with those of the bi-bimodule model. The proposed generalization concerns all higher dimensions and all morphism levels.

The conjecture is motivated by the recursive identification of algebras in bimodule categories with iterated bimodule structures. Establishing it would show that the three higher Morita formalisms agree throughout the entire hierarchy rather than only in the dimensions and levels treated explicitly.

References

We propose that the category of bi-bi-bimodules is equivalent to the category of bimodules internalized to bi-bi-modules and this equivalence also holds for $n=4,5,\dots$ cases. Thus for any $n$, there should be

— 2-Morita Theory of $E_2$-Algebras and Module Categories  (2608.27228 - Xu et al., 27 Aug 2026) in Section 4, immediately after the discussion of the n-Morita equivalences and the preceding n=2 result