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Correspondence between Theriault’s dual polyhedral product and the polyhedral coproduct

Determine whether there exists any general correspondence between Theriault’s dual polyhedral product (X,*)^K_D and the polyhedral coproduct (X,*)^K_co beyond the two known cases where K consists of m disjoint vertices or K equals the boundary of the (m−1)-simplex ∂Δ^{m−1}.

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Background

The paper compares two constructions: Theriault’s dual polyhedral product (X,)K_D and the new polyhedral coproduct (X,)K_co defined as a homotopy limit of wedges over the face poset of K.

They coincide in two specific cases: when K is m disjoint points (both recover the thin/product extremes) and when K = ∂Δ{m−1} (both yield the cartesian product). Beyond these, the authors note that the diagrammatic definitions differ substantially and any broader relationship is unclear.

References

In particular, when K is m disjoint points, (X,)K_D is equal to the thin product of X_1,\ldots,X_m. When K = \partial{\Delta{m-1}, (X,)K_D \simeq X_1 \times \cdots \times X_m. Outside of these cases, it is not clear whether there is any correspondence between the dual polyhedral product, and the polyhedral coproduct.

— Polyhedral coproducts (2405.19258 - Amelotte et al., 29 May 2024) in Remark (rem:coprodvsdual), Section 2.1 (Basic examples)