ScalingStacks

Proof.

We prove the first statement; the proof of the second statement is analogous. Suppose \(R\) preserves filtered colimits and \(c\in \mathcal C\) is compact. Then, for every filtered diagram \(d\colon I \rightarrow\mathcal D\) we have \[\begin{gathered} \mathrm{Hom}_{\mathcal C}(Lc, \mathrm{colim}_i d_i) \simeq \mathrm{Hom}_{\mathcal D}(c, R\mathrm{colim}_i d_i) \simeq \mathrm{Hom}_{\mathcal D}(c,\mathrm{colim}_i Rd_i) \\ \hspace{3cm}\simeq \mathrm{colim}_i \mathrm{Hom}_{\mathcal D}(c, Rd_i) \simeq \mathrm{colim}_i \mathrm{Hom}_{\mathcal D}(Lc, d_i) \end{gathered}\] and hence \(Lc\) is compact. Conversely, suppose that \(L\) preserves compact objects. It follows that for a compact object \(c\in \mathcal C\) and a filtered diagram \(d\colon I \rightarrow\mathcal D\), we have \[\begin{gathered} \mathrm{Hom}_{\mathcal C}(c, R(\mathrm{colim}_i d_i))\simeq \mathrm{Hom}_{\mathcal D}(Lc, \mathrm{colim}_i d_i) \simeq \mathrm{colim}_i \mathrm{Hom}_{\mathcal D}(Lc, d_i) \\ \hspace{3cm} \simeq \mathrm{colim}_i \mathrm{Hom}_{\mathcal C}(c, Rd_i)\simeq \mathrm{Hom}_{\mathcal C}(c, \mathrm{colim}_i Rd_i). \end{gathered}\] Since \(\mathcal C\) is compactly generated, every object in \(\mathcal C\) is a small colimit of compact objects, and thus \(\mathrm{colim}_i Rd_i \simeq R\mathrm{colim}_i d_i\). ◻