The classifying topos for infinity-connected objects

Every object \(X \in T\) determines a morphism of logoi \(\PSh(\An^{\fin,\op}) \to T\) given on generators by \(A \mapsto X^A\). The object \(X\) is \(\infty\)-connected if and only if, for every morphism \(f\colon A\to B\) of finite animae, the induced map is an effective epimorphism. One direction follows by building finite animae from finitely many cells and using stability of effective epimorphisms under base change and composition. Conversely, the maps between finite spheres include the tests asserting that every iterated diagonal of \(X\to *\) is an effective epimorphism. This criterion is also recorded in [Anel et al. 2025, Example 2.1.16(g)].Let \(K\) be the kernel of \(\colim\colon\Fun(\An^{\fin},\An)\to\An\). By Lemma 5.56, its monogenic part is generated by the monomorphisms \(\im(y(f))\) for maps \(f\) of finite animae. A morphism of logoi preserves epi–mono factorizations, so the morphism classified by \(X\) sends \(\im(y(f))\) to an isomorphism if and only if it sends \(y(f)\) to an effective epimorphism, which is precisely the condition above. Thus the free morphism classified by \(X\) factors through \(\Fun(\An^{\fin},\An)/K^{\mono}\) exactly when \(X\) is \(\infty\)-connected. The universal property of the free logos therefore restricts from to the desired equivalence