Plethysms and operads
| Autor: | Cebrian, Alex |
|---|---|
| Rok vydání: | 2023 |
| Předmět: |
Mathematics::K-Theory and Homology
Mathematics::Category Theory Applied Mathematics General Mathematics FOS: Mathematics Mathematics - Combinatorics Algebraic Topology (math.AT) Mathematics - Category Theory Category Theory (math.CT) 05A19 18B40 18N50 16T10 18M65 18M80 13J05 Mathematics - Algebraic Topology Combinatorics (math.CO) Mathematics::Algebraic Topology |
| Zdroj: | Collectanea Mathematica. |
| ISSN: | 2038-4815 0010-0757 |
| DOI: | 10.1007/s13348-022-00386-1 |
| Popis: | 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. Comment: 68 pages, expository improvements and minor corrections |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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