The Symmetric Join Operad

TL;DR

This paper develops a symmetric version of the join operad, incorporating permutation symmetries, and constructs an E-infinity operad that acts on chains of simplicial sets, advancing the understanding of operadic symmetries in algebraic topology.

Contribution

It introduces a simplicial symmetric join operad and constructs an E-infinity operad with natural coaction on simplicial chains, enhancing operadic symmetry understanding.

Findings

Constructed a symmetric join operad incorporating permutation symmetries.

Developed an E-infinity operad that coacts on chains of simplicial sets.

Provides a combinatorial framework for operadic symmetries in topology.

Abstract

The join operad arises from the combinatorial study of the iterated join of simplices. We study a suitable simplicial version of this operad which includes the symmetries given by permutations of the factors of the join. From this combinatorics we construct an E-infinity operad which coacts naturally on the chains of a simplicial set.