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 $j \colon Δ_{\operatorname{inert}} \to Δ$ takes general objects to Möbius decomposition spaces and general maps to CULF maps. We establish an equivalence of $\infty$-categories $\mathbf{PrSh}(Δ_{\operatorname{inert}}) \simeq \mathbf{Decomp}_{/B\mathbb{N}}$. 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.

31 pages. v2: Accepted version. Many improvements based on suggestions of the referee. Added a proof that j_! is fully faithful. Streamlined applications sections. Removed discussion of IKEO maps, Aguiar--Bergeron--Sottile map, and zeta functions -- this material will be given an expanded treatment elsewhere