We introduce the concept of a free decomposition space. A free decomposition space is a simplicial space that is freely generated by inert maps. Left Kan expansion along containment $j \colon \Delta_{\operatorname{inert}} \to \Delta$ indicates taking a general object into the M\”obius decomposition space and transforming the general map into a CULF map. Establish equality of $ \infty$-categories $\mathbf{PrSh}(\Delta_{\operatorname{inert}}) \simeq \mathbf{Decomp}_{/B\mathbb{N}} $. Although free decomposition spaces are fairly simple objects, they are rich in combinatorics. All separable square products appear to arise from free decomposition spaces. Extensive list of examples involving quasisymmetric functions We show that Aguiar–Bergeron-Sottile maps to the decomposition space We factorize the decomposition space and provide a conceptual description of the zeta function contained in the universal property of $\operatorname{. QSym}$.

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