Inert active factorization

Remark 14.1.12. Let \(\Oo = (\Oo ^{\otimes },p_{\Oo })\) be an \(\infty \)-operad. Then every morphism \(\phi \colon X \to Z\) in \(\Oo ^{\otimes }\) factors uniquely as \[ X \xrightarrow {\widetilde {f}} Y \xrightarrow {\psi } Z, \] where \(\widetilde {f}\) is an inert morphism and \(\psi \) is an active morphism. To see this, let us denote by \(I \xleftarrow {f} K \xrightarrow {g} J\) the image of \(\phi \) under \(p\). This span may be factored uniquely as a composite of a backwards span \(I \xleftarrow {f} K \xrightarrow {=} K\) followed by a forward span \(K \xleftarrow {=} K \xrightarrow {g} J\). We then let \(\widetilde {f}\colon X \to Y\) be the unique cocartesian lift of \(f\) with domain \(X\). It follows that there is a unique morphism \(\psi \colon Y \to Z\) over \(g\) such that \(\psi \circ \widetilde {f} = \phi \).

The argument above produces a factorization and shows that any two of them are isomorphic. With more care, it in fact shows that the factorization is unique, in the sense that the anima of factorizations of \(\phi \) is contractible. Rather than spelling this out by hand, we will obtain it in Section 17.1 from the general theory of factorization systems on an \(\infty \)-category, where we show that the inert and active morphisms define such a system on \(\Oo ^{\otimes }\).