Plethysms and operads

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.