Plethysms and operads

TL;DR

This paper introduces a new $ ext{T}$-construction on generalized operads that unifies various notions of plethystic substitution, connecting operad theory with combinatorial and homotopical structures.

Contribution

It defines the $ ext{T}$-construction as a general framework for plethystic substitution, recovering known constructions and providing new combinatorial models for plethysm.

Findings

Realizes plethysm as convolution products from homotopy cardinality.

Recovers Giraudo's $T$-construction for monoids.

Provides a combinatorial model for ordinary plethysm.

Abstract

We introduce the $\mathcal{T}$-construction, an endofunctor on the category of generalized operads as a general mechanism by which various notions of plethystic substitution arise from more ordinary notions of substitution. In the special case of one-object unary operads, i.e. monoids, we recover the $T$-construction of Giraudo. We realize several kinds of plethysm as convolution products arising from the homotopy cardinality of the incidence bialgebra of the bar construction of various operads obtained from the $\mathcal{T}$-construction. The bar constructions are simplicial groupoids, and in the special case of the terminal reduced operad $\mathsf{Sym}$, we recover the simplicial groupoid of arXiv:1804.09462, a combinatorial model for ordinary plethysm in the sense of P\'olya, given in the spirit of Waldhausen $S$ and Quillen $Q$ constructions. In some of the cases of the $\mathcal{T}$-construction, an analogous interpretation is possible.