If \(L\) is an acyclic class, then \(D(L)\) is again an acyclic class. If \(L\) is of small generation, then so is \(D(L)\); in this case both classes are left classes of modalities.
Proof
Stability under base change and the inclusion of all isomorphisms are immediate. For composition, let \(A \xrightarrow{f} B \xrightarrow{g} C\) lie in \(D(L)\). Then \(gf\in L\), and \(\Delta_{gf}\) factors as
\[A \xrightarrow{ \Delta_f } A \times_B A \longrightarrow A \times_C A ,\]
where the second map is a base change of \(\Delta_g\). Since \(\Delta_f,\Delta_g\in L\) and \(L\) is an acyclic class, \(\Delta_{gf}\in L\).For colimits, let \(f_{\bullet}\colon X_{\bullet}\to Y_{\bullet}\) be a diagram in \(\Ar(T)\) with every \(f_i\in D(L)\), and write \(f\colon X\to Y\) for its colimit. We already know \(f\in L\). The acyclic-descent argument of [Anel et al. 2024, Section 2.2.7] says that colimits of \(L\)-cartesian transformations remain \(L\)-cartesian. We spell out its application here, since it also fixes the variance of the maps involved.For a morphism \(i\to j\) in the indexing category, the gap map of the naturality square is
The first map is a base change of \(\Delta_{f_j}\), and the second is a base change of \(f_i\). Both lie in \(L\), so the transformation \(f_{\bullet}\) is \(L\)-cartesian. Acyclic descent now shows that for every \(i\) the square is \(L\)-cartesian. Equivalently, the gap map \(h_i\colon X_i\to Y_i\times_YX\) lies in \(L\).By universality of colimits, the diagonal \(\Delta_f\) is the colimit of the maps \(X_i\to X_i\times_YX\). Each of these factors as
where the second map is a base change of \(h_i\). Thus both maps lie in \(L\). Closure under colimits gives \(\Delta_f\in L\), proving that \(D(L)\) is saturated. The small-generation assertion is the accessibility part of the cited décalage result.
References
Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of ∞-topoi. II: Grothendieck topologies. J. Pure Appl. Algebra, 228 (3), 63. 2024.
