Consequences of complementary classes

For (1), take \(C=\Pp^{K'}(D)\) in the definition of complementarity. The relative completion \(\Pp^{\all}_{K'}(\Pp^{K'}(D))\) is \(\Pp^{\all}(D)\) by the two universal properties. Hence every presheaf on \(D\) is obtained from representables by first forming \(K'\)-indexed colimits and then \(K\)-indexed colimits. In particular, the subcategory generated under both classes is all of \(\PSh(D)\).For (2), let \(I \in K'\), \(J \in K\), and let \(X\colon I \times J \to \An\) be a diagram. We must show that the canonical map \(\colim_{j \in J}\lim_{i \in I} X_{i,j} \to \lim_{i \in I}\colim_{j \in J} X_{i,j}\) is an isomorphism. Let \(C := \PSh(I\catop)\), and for each \(j \in J\) let \(d_j \in C\) be the presheaf \(i \mapsto X_{i,j}\). Also let By the large-\(C\) variant of Remark 6.180 (using small presheaves), \(\Pp^{\all}_{K'}(C)\) is a full subcategory of \(\PSh(C)\). Using the assumption \(K \perp K'\), this full subcategory agrees with \(\Pp^K(C) \subseteq \PSh(C)\), hence is closed under \(K\)-indexed colimits. Since each \(y(d_j)\) belongs to it, it follows that \(F \in \Pp^{\all}_{K'}(C)\), and thus \(F\) sends the \(I\catop\)-indexed colimit \(u=\colim_{i \in I\catop} y(i)\) in \(C\) to an \(I\)-indexed limit in \(\An\): Now compute both sides using pointwise colimits and Yoneda: while This proves (2).