The relative join operad and polyhedral products
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.
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BibTeX
@misc{vallee2026relative,
title = {{The relative join operad and polyhedral products}},
author = {Mathieu Vallée},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.02440},
eprint = {2605.02440},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.02440},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The relative join operad and polyhedral products
AU - Mathieu Vallée
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2605.02440
UR - https://arxiv.org/abs/2605.02440
AB - 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.
ER -