Directed hereditary species and decomposition spaces
Abstract
We introduce the notion of directed hereditary species and show that they have associated monoidal decomposition spaces, comodule bialgebras, and operadic categories. The notion subsumes Schmitt's hereditary species, G\'alvez--Kock--Tonks directed restrictions species, and a directed version of Carlier's construction of monoidal decomposition spaces and comodule bialgebras. In addition to all the examples of Schmitt, G\'alvez--Kock--Tonks and Carlier, the new construction covers also the Fauvet--Foissy--Manchon comodule bialgebra of finite topological spaces, the Calaque--Ebrahimi-Fard--Manchon comodule bialgebra of rooted trees, and the Fa\`a di Bruno comodule bialgebra of linear trees.
Cite
@article{arxiv.2211.07721,
title = {Directed hereditary species and decomposition spaces},
author = {Alex Cebrian and Wilson Forero},
journal= {arXiv preprint arXiv:2211.07721},
year = {2023}
}
Comments
Many results have been rewritten to make them easier to understand. The arguments used in subsection 7.5 are similar to those given in Section 5 arXiv:1903.07964 considering contractions instead of monotone surjections. We prefer to add the proof to make the paper as self-contained as possible, but in any case, the ideas come from Carlier