
TL;DR
This paper introduces a method to construct set-theoretic operads using the power set functor, revealing a hierarchy connecting classical operads and explaining polyhedral product constructions.
Contribution
It systematically develops a hierarchy of operads via power set iterations, linking classical operads and polyhedral product structures in a unified framework.
Findings
The first iteration recovers the commutative triassociative operad.
The second iteration yields the substitution and composition operads on simplicial complexes.
Both the substitution and composition operads are infinitely generated.
Abstract
We introduce a systematic method for constructing set-theoretic operads via iterated application of the power set functor, and use it to uncover a hierarchy connecting several classical operads. Starting from the permutative operad, the first iteration recovers the commutative triassociative operad. The second iteration produces the substitution operad and the composition operad on simplicial complexes, two structures introduced by Ayzenberg and Abramyan--Panov in the theory of polyhedral products; we prove that both are infinitely generated. This hierarchy yields a conceptual explanation for the multiplicity of polyhedral product constructions: the arrows of any cocontinuous cocomplete symmetric monoidal category carry natural algebra structures over both operads, recovering the Cartesian, smash, and join polyhedral products as instances for different monoidal structures on…
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