  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.
