Compactness coherent objects

Pulling a filtered diagram back to the slice \(T_{/X}\) identifies the required mapping anima with the anima of sections. We may therefore replace \(T\) by \(T_{/X}\) and assume that \(X=*\); the hypothesis becomes the corresponding coherence condition on the terminal object of the slice. We prove the claim by induction on \(n\). For \(n = 0\), the assertion is exactly the definition of quasi-compactness, expressed in terms of filtered unions of subterminal objects.Now consider a filtered category \(I\) and an object \(Y \in \Fun(I, T_{\leq n-1})\). We have to show that the canonical comparison map is an isomorphism. We will show the following claim:Claim. The map \(\beta_Y\) is an effective epimorphism.Assuming this for a moment, we may conclude. It remains to check that \(\beta_Y\) induces isomorphisms on path animae. Consider two points \(x,y \in \colim_i \Hom(X,Y_i)\). We may represent both at a common stage and, after replacing \(I\) by the corresponding undercategory, assume that this stage is an initial object \(0 \in I\). For each \(i\), let where the second map to \(Y_i\times Y_i\) is the diagonal. The anima of sections of \(E_i\to X\) is the path anima from \(x_i\) to \(y_i\). Since \(Y_i\) is \((n-1)\)-truncated, \(E_i\to X\) is \((n-2)\)-truncated. Filtered colimits commute with finite limits in a topos, so the analogous object for \(\colim_iY_i\) is \(\colim_iE_i\). The induction hypothesis in \(T_{/X}\) therefore identifies the two path animae.Let us now prove the claim. Given a map \(X \to \colim_i Y_i\), we need to show that it factors through some finite stage \(Y_i \to \colim_i Y_i\). Since \(X\) is quasi-compact and locally \((n-1)\)-coherent, there is an effective epimorphism \(U \twoheadrightarrow X\) where \(U\) is \((n-1)\)-coherent, satisfying the additional property that the composite \(U \twoheadrightarrow X \to \colim_i Y_i\) factors through some \(Y_0\), where \(0 \in I\). (This uses that \(I\) is filtered.) Letting \(U_{\bullet} := \check{C}_{\bullet}(U \to X)\) and \(V_{i,\bullet} := \check{C}_{\bullet}(Y_0 \to Y_i)\), then the commutative diagram

induces a map on Čech nerves of the form Since \(U\) is \((n-1)\)-coherent and \(X\) is \(n\)-coherent, each \(U_m\) is \((n-1)\)-coherent.By induction on \(m \leq n\), we show that the map \(\sk_m(U_{\bullet}) \to \colim_i \sk_m(V_{i,\bullet})\) factors through some \(\sk_m(V_{i,\bullet})\), as semisimplicial objects. Here \(\sk_m\) denotes the \(m\)-skeleton. At the successor step, the boundary of the desired \(m\)-simplex has already been chosen at a finite stage. The remaining extension problem is a problem of sections for the pullback of This map is \((n-m-1)\)-truncated, while \(U_m\) is \((n-1)\)-coherent; the induction hypothesis in the relevant slice therefore moves the extension to a finite stage. Filteredness lets us choose one stage for the finitely many faces and degeneracies. With \(m=n\), we obtain and since \(\Hom_T(X,Y_i) \cong \Hom_T(\tau_{n-1}(X), Y_i)\) we may conclude.This is the skeletal lifting argument of [Lurie 2018, Proposition A.2.3.1]; the preceding discussion records the coherence and truncation estimates needed in its application here.