Finite dimension implies Postnikov properties

(1) Let \(\{U_\alpha\}\) be a collection of generators such that every slice \(T_{/U_\alpha}\) has finite dimension, say at most \(d_\alpha\). For every \(X \in T\), the dimension bound shows that is \((n-d_\alpha)\)-connected. The target is \(n\)-truncated. The standard convergence criterion for towers of animae therefore identifies \(\Hom_T(U_\alpha,X)\) with \(\lim_n\Hom_T(U_\alpha,\tau_nX)\): in each fixed degree, the homotopy groups and the transition maps stabilize. This is also the mapping-anima argument in the proof of [Lurie 2009, Proposition 7.2.1.10]. Since the \(U_\alpha\) generate \(T\), it follows that \(X \to \lim_n\tau_nX\) is an isomorphism.If \(f\colon X\to Y\) is \(\infty\)-connected, then \(\tau_nf\) is an isomorphism for every \(n\). Applying the convergence statement to \(X\) and \(Y\) shows that \(f\) itself is an isomorphism. Hence \(T\) is hypercomplete, in agreement with [Lurie 2009, Corollary 7.2.1.12].(2) By (1), the comparison \(T \to \widehat T\) is fully faithful. Let \((X_n)_n \in \widehat T\) and put \(X:=\lim_nX_n\). The transition map \(X_{m+1}\to X_m\iso\tau_mX_{m+1}\) is \(m\)-connected. The highly connected tower criterion proved in [Lurie 2009, Proposition 7.2.1.10], applied with the uniform bound \(d\), shows that, for every \(n\), the map \(X\to X_m\) is \(n\)-connected for all sufficiently large \(m\). For \(m\geq n\), the composite is therefore \(n\)-connected. Since \(X_n\) is \(n\)-truncated, this identifies \(X_n\) with \(\tau_nX\). Thus every object of \(\widehat T\) is the Postnikov tower of an object of \(T\), so \(T\to\widehat T\) is essentially surjective.