Power set operads

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 topological spaces. Going further, we construct a new operad on relative simplicial complexes, governed by the join polyhedral product, which contains both the composition and the substitution operads as suboperads. As an application, pairs of piecewise-linear balls without interior vertices with their boundary spheres form a suboperad, extending the stability of the $J$-construction on piecewise-linear~spheres.