Free decomposition spaces

Abstract

We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by inert maps. We show that left Kan extension along the inclusion takes general objects to M & ouml;bius decomposition spaces and general maps to CULF maps. We establish an equivalence of infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-categories . 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.