ScalingStacks

Proof.

The first statement is [Lur17, Lem. 5.3.2.9] for \(\kappa=\omega\), also see [Lur09, Prop. 5.5.7.8].

To prove the second statement, we note that [Lur09, Cor. 5.3.6.10, Rem. 5.5.8.16] implies that \(\mathcal P^{\Sigma}(-)\) defines a functor from \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) to the huge \(\infty\)-category \(\widehat{\mathrm{Cat}}_{\infty}^{\mathrm{cocpl}}\) of large \(\infty\)-categories which admit all small colimits and colimit preserving functors. By [Lur09, Prop. 5.5.8.10], \(\mathcal P^{\Sigma}(\mathcal C)\) is an accessible localization of \(\mathcal P(\mathcal C)\) and hence is presentable, so that \(\mathcal P^{\Sigma}\) factors through the full subcategory \(\mathrm{Pr}^\mathrm{L}\) of \(\widehat{\mathrm{Cat}}_{\infty}^{\mathrm{cocpl}}\). By [Lur09, Prop. 5.5.8.22], every object in the image of the Yoneda embedding \(\mathcal C\hookrightarrow \mathcal P^{\Sigma}(\mathcal C)\) is compact-projective, and since \(\mathcal P^{\Sigma}(\mathcal C)\) is a localization of \(\mathcal P(\mathcal C)\), it is generated under small colimits by objects in \(\mathcal C\); hence \(\mathcal P^{\Sigma}(\mathcal C)\) is projectively generated. Moreover, by [Lur09, Prop. 5.5.8.25], the compact-projective objects of \(\mathcal P^{\Sigma}(\mathcal C)\) are precisely the objects in (the essential image of) \(\mathcal C\) (note: this uses that \(\mathcal C\) is idempotent complete). Therefore, for any morphism \(f\colon \mathcal C\rightarrow\mathcal D\) in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) the cocontinuous functor \(\mathcal P^{\Sigma}(f)\colon \mathcal P^{\Sigma}(\mathcal C) \rightarrow\mathcal P^{\Sigma}(\mathcal D)\) preserves compact-projectives; hence \(\mathcal P^{\Sigma}\colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Pr}^\mathrm{L}\) factors through the subcategory \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow \mathrm{Pr}^\mathrm{L}\). To show that it factors as an equivalence, notice that it is fully faithful since for \(\mathcal C, \mathcal D\in \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), \[\mathrm{Fun}^{\sqcup}(\mathcal C, \mathcal D) \simeq \mathrm{Fun}^{\sqcup}(\mathcal C, \mathcal P^{\Sigma}(\mathcal D)^{\mathrm{cp}}) \simeq\mathrm{\mathrm{Fun}^{L, cp}} (\mathcal P^{\Sigma}(\mathcal C), \mathcal P^{\Sigma}(\mathcal D))\] where the first equivalence uses that \(\mathcal D\simeq \mathcal P^{\Sigma}(\mathcal D)^{\mathrm{cp}}\) and the second equivalence uses that for any presentable \(\mathcal E\), the map \(\mathrm{Fun^L}(\mathcal P^{\Sigma}(\mathcal C), \mathcal E) \rightarrow\mathrm{Fun}^{\sqcup}(\mathcal C, \mathcal E)\) is an isomorphism (this follows e.g. from [Lur09, Prop. 5.5.8.10]) and the fact that \(\mathcal C\simeq \mathcal P^{\Sigma}(\mathcal C)^{\mathrm{cp}}\). Lastly, surjectivity on objects follows since by [Lur09, Prop. 5.5.8.25] any projectively generated presentable \(\infty\)-category \(\mathcal D\) is equivalent to \(\mathcal P^{\Sigma}(\mathcal C)\) where \(\mathcal C\) is the smallest full subcategory of \(\mathcal D\) spanned by finite coproducts of objects in the set \(S\) of compact-projective generators.

The third statement follows since the induced composite \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) sends \(\mathcal C\) to \(\mathcal P^{\Sigma}(\mathcal C)^{\mathrm{c}}\), which in the notation of proposition 3.1.11 is equivalent to \(\mathcal P_{\mathcal K}^{\mathcal K'}(\mathcal C)\) for \(\mathcal K\) the collection of finite sets together with the ‘walking idempotent’ \(\mathrm{Idem}\) of [Lur09, Sec. 4.4.5] and \(\mathcal K'\) the collection of finite categories together with \(\mathrm{Idem}\). Hence, by proposition 3.1.11.([003D]) this functor \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) is left adjoint to the forgetful functor \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\rightarrow \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\). The fourth statement then follows from the previous ones and proposition 3.1.11.([003C]), since \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) and \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) are presentable and since the functor \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) is equivalent to \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) and hence a left adjoint. ◻

Proof.

The first statement is [Lur09, Prop. 5.5.8.25.(2).(iii)], the proof of the second statement is analogous. ◻

Proof.

Recall from proposition 3.1.11 the functor \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) which is left adjoint to the forgetful functor. By proposition 3.2.8.([003Z]), this functor is equivalent to the composite \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\simeq \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\simeq \mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) for the subcategory inclusion \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\). Hence, this induces presentably symmetric monoidal structures on \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) and \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) and on the inclusion \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\).

It remains to show that the subcategory inclusion \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\rightarrow\mathrm{Pr}^\mathrm{L}\) is symmetric monoidal which follows since the composite \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\xrightarrow{\operatorname{Ind}} \mathrm{Pr}^{\mathrm{L},\mathrm{c}}\rightarrow \mathrm{Pr}^\mathrm{L}\) is [Lur17, Lem. 5.3.2.11]. ◻

Proof.

For any small \(\infty\)-category \(\mathcal C\), the presheaf category \(\mathcal P(\mathcal C)\) is generated by a small set of tiny objects, i.e. objects \(c\in \mathcal C\) for which \(\mathrm{Hom}_{\mathcal C}(c,-)\colon \mathcal C\rightarrow\mathcal S\) preserves all small colimits; such a small set of tiny objects is for example provided by the objects of \(\mathcal C\) itself. Moreover, any functor \(F\colon \mathcal C\rightarrow\mathcal D\) induces a left adjoint functor \(\mathcal P(F)\colon \mathcal P(\mathcal C) \rightarrow\mathcal P(\mathcal D)\) which preserves tiny objects. By an argument entirely analogous to the proof of corollary 3.2.6, it follows that \(\mathcal P(\mathcal C)\) is in particular projectively generated, and that \(\mathcal P(F)\) preserves compact-projectives. Hence, the symmetric monoidal functor \(\mathcal P\colon \mathrm{Cat}_{\infty}\rightarrow\mathrm{Pr}^\mathrm{L}\) factors through the symmetric monoidal subcategory \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) from proposition 3.2.10. Moreover, the functor \(\mathrm{Cat}_{\infty}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) is a left adjoint since after composing with the equivalence \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\simeq \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) from proposition 3.2.8 it becomes equivalent to the left adjoint \(\mathrm{Cat}_{\infty}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) of the forgetful functor. Since moreover \(\mathcal S\rightarrow\mathrm{Cat}_{\infty}\) is a left adjoint, so is the composite \(\mathcal S\rightarrow\mathrm{Cat}_{\infty}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\). ◻

Proof.

Statements ([004C]) to ([004E]) are proven in [Lur17, Cor. 7.1.4.14], statement ([004F]) is an immediate consequence. Statements ([004G]) to ([004J]) can be proven analogously (and also follow from the proof of [Lur17, Lem. 5.3.2.12 (3)]). ◻