The relative join operad and polyhedral products

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Abstract

An inclusion of nonvoid simplicial complexes induces an arrow of polyhedral products. Applying Ayzenberg’s polyhedral join to both complexes gives a symmetric relative join operad. Its endpoint suboperads recover Abramyan–Panov substitution and Ayzenberg composition. Neither endpoint operad nor either relative join operad is finitely generated. The suboperad with nonvoid lower complex arises as the nonempty power-set quotient of the subset-inclusion operad. When tensoring preserves colimits of nonempty finite diagrams, the suboperad acts on arrows by polyhedral colimits up to coherent natural isomorphism. The action induces a set-operad algebra on isomorphism classes of arrows. The dual construction produces Stanley–Reisner quotient arrows, and the colimit action refines Eldridge’s loop-space decomposition to arrows. PL ball–boundary pairs form a suboperad of the relative join operad, and minimal interior faces give an operad morphism. Principal pairs have odd-dimensional spherical moment-angle homotopy fibers and yield a closure result for Eldridge’s loop-space class.

Citation

Vallée, Mathieu. 2026. “The relative join operad and polyhedral products.” https://arxiv.org/abs/2605.02440

@misc{vallee_relative_2026,
  author = {Vallée, Mathieu},
  year = {2026},
  title = {The relative join operad and polyhedral products},
  doi = {10.48550/arXiv.2605.02440},
  url = {http://arxiv.org/abs/2605.02440},
  note = {arXiv:2605.02440 [math.AT]},
  publisher = {arXiv},
}