Plethysms and operads [article]

Alex Cebrian
2020 arXiv   pre-print
We introduce the 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
more » ... the 𝒯-construction. The bar constructions are simplicial groupoids, and in the special case of the terminal reduced operad 𝖲𝗒𝗆, we recover the simplicial groupoid of arXiv:1804.09462, a combinatorial model for ordinary plethysm in the sense of Pólya, given in the spirit of Waldhausen S and Quillen Q constructions. In some of the cases of the 𝒯-construction, an analogous interpretation is possible.
arXiv:2008.09798v2 fatcat:nfixhhk3wze2jk2m365mzkdiym