# Hereditary species as monoidal decomposition spaces, comodule bialgebras, and operadic categories

## TL;DR

This paper demonstrates how hereditary species lead to monoidal decomposition spaces, resulting in new bialgebra and operadic category structures, linking combinatorial species with advanced algebraic frameworks.

## Contribution

It establishes a novel connection between hereditary species, monoidal decomposition spaces, and operadic categories, expanding the algebraic understanding of combinatorial structures.

## Key findings

- Hereditary species induce monoidal decomposition spaces.
- Schmitt's bialgebra construction is an instance of a general bialgebra on these spaces.
- Hereditary species produce new examples of operadic categories.

## Abstract

We show that Schmitt's hereditary species induce monoidal decomposition spaces, and exhibit Schmitt's bialgebra construction as an instance of the general bialgebra construction on a monoidal decomposition space. We show furthermore that this bialgebra structure coacts on the underlying restriction-species bialgebra structure so as to form a comodule bialgebra. Finally, we show that hereditary species induce a new family of examples of operadic categories in the sense of Batanin and Markl.