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 Möbius decomposition spaces and general maps to CULF maps. We establish an equivalence of \(\infty \) -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.
