Free decomposition spaces

TL;DR

This paper introduces free decomposition spaces, a new class of simplicial spaces generated by inert maps, and explores their properties and applications in combinatorics, including connections to quasi-symmetric functions.

Contribution

It defines free decomposition spaces, shows their relation to M"obius decomposition spaces via Kan extension, and establishes an equivalence of categories, with applications to combinatorics.

Findings

Free decomposition spaces are generated by inert maps.

Left Kan extension maps objects to M"obius decomposition spaces.

All deconcatenation type comultiplications arise from free decomposition spaces.

Abstract

We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by their inert maps. We show that left Kan extension along the inclusion $j \colon \Delta_{\operatorname{inert}} \to \Delta$ takes general objects to M\"obius decomposition spaces and general maps to CULF maps. We establish an equivalence of $\infty$-categories $\mathbf{PrSh}(\Delta_{\operatorname{inert}}) \simeq \mathbf{Decomp}_{/B\mathbb{N}}$. Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions.