Proposition 3.3.4 (004R) · ScalingStacks

The following hold.

The equivalence \(\mathcal P^{\Sigma}\colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) restricts to an equivalence between full subcategories \[\mathcal P^{\Sigma}\colon\mathrm{add}\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}.\] Its inverse is \((-)^{\mathrm{cp}}\) which takes a projectively generated additive presentable \(\infty\)-category \(\mathcal C\) to its full subcategory \(\mathcal C^{\mathrm{cp}}\) on the compact-projective objects.
The equivalence \(\operatorname{Ind}\colon \mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) restricts to an equivalence between full subcategories \[\operatorname{Ind}\colon\mathrm{st}\xrightarrow{\simeq}\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}.\] Its inverse is \((-)^{\mathrm{c}}\) which takes a compactly generated stable presentable \(\infty\)-category \(\mathcal C\) to its full subcategory \(\mathcal C^{\mathrm{c}}\) on the compact objects.

Proof.

We will prove part ([004S]), the proof of part ([004T]) is entirely analogous and can for example be found in [BGT13, Lem. 2.20]. Recall from proposition 3.2.8 that \(\mathcal P^{\Sigma}\colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) is an equivalence, whose inverse is \((-)^{\mathrm{cp}}\). To prove statement (1), it therefore suffices to show that the essential image of the composite \(\mathrm{add}\hookrightarrow \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\simeq \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) is the full subcategory \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\).

If \(\mathcal C\) is a small additive \(\infty\)-category, then \(\mathcal P^{\Sigma}(\mathcal C) \simeq \mathrm{Fun}^{\sqcup}(\mathcal C^{\mathrm{op}}, \mathcal S)\) is additive by [GGN15, Cor. 2.9]. On the other hand, if \(\mathcal D\) is any projectively generated additive presentable category, then the full subcategory on its compact-projective objects is closed under finite coproducts and hence is again additive. Therefore, \(\mathcal D\) is in the image of \(\mathrm{add}\hookrightarrow \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\). ◻