Directed hereditary species and decomposition spaces

TL;DR

This paper introduces directed hereditary species, unifying various algebraic structures and constructions into a comprehensive framework involving monoidal decomposition spaces and comodule bialgebras.

Contribution

It defines directed hereditary species and demonstrates their connection to monoidal decomposition spaces, unifying multiple existing algebraic structures and examples.

Findings

Unified framework for hereditary species and related structures.

Construction of new comodule bialgebras from directed hereditary species.

Covers classical and modern algebraic structures like rooted trees and finite topological 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.