Free decomposition spaces

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 $j\text{\hspace{0.17em}⁣}:{\mathrm{\Delta }}_{\mathrm{inert}}\to \mathrm{\Delta }$ takes general objects to Möbius decomposition spaces and general maps to CULF maps. We establish an equivalence of $\mathrm{\infty }$-categories $\mathbf{P}\mathbf{r}\mathbf{S}\mathbf{h}({\mathrm{\Delta }}_{\mathrm{inert}})\simeq {\mathbf{D}\mathbf{e}\mathbf{c}\mathbf{o}\mathbf{m}\mathbf{p}}_{\mathrm{/}B\mathbb{N}}$. 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.