9.7.8 Idempotent $\infty $-Categories
Let $n$ be an integer, and let $\operatorname{\mathcal{S}}^{\leq n}$ denote the $\infty $-category of $n$-truncated spaces. For every presentable $\infty $-category $\operatorname{\mathcal{C}}$, the cocomplete tensor product $\operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{C}}$ can be identified with the full subcategory $\operatorname{\mathcal{C}}^{\leq n}$ spanned by the $n$-truncated objects of $\operatorname{\mathcal{C}}$ (Proposition 9.7.6.5). Note that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\leq n}$ is idempotent: if $\operatorname{\mathcal{C}}$ is locally $n$-truncated (for example, if $\operatorname{\mathcal{C}}= \operatorname{\mathcal{D}}^{\leq n}$ for some presentable $\infty $-category $\operatorname{\mathcal{D}}$), then $\operatorname{\mathcal{C}}^{\leq n}$ coincides with $\operatorname{\mathcal{C}}$. We now place this phenomenon in an axiomatic framework.
Definition 9.7.8.1. Let $\operatorname{\mathcal{E}}$ be a presentable $\infty $-category. We say that $\operatorname{\mathcal{E}}$ is a presentable idempotent if there exists an object $E \in \operatorname{\mathcal{E}}$ for which the functor
\[ \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\widehat{\otimes } \operatorname{\mathcal{E}}\quad \quad X \mapsto E \otimes X \]
is an equivalence of $\infty $-categories; here $\operatorname{\mathcal{E}}\widehat{\otimes } \operatorname{\mathcal{E}}$ denotes the cocomplete tensor product of Variant 9.7.3.15. In this case, we say that the object $E \in \operatorname{\mathcal{E}}$ exhibits $\operatorname{\mathcal{E}}$ as a presentable idempotent.
Example 9.7.8.2. For every integer $n$, the $\infty $-category $\operatorname{\mathcal{S}}^{\leq n}$ of $n$-truncated Kan complexes is a presentable idempotent. More precisely, the object $\Delta ^0 \in \operatorname{\mathcal{S}}^{\leq n}$ exhibits $\operatorname{\mathcal{S}}^{\leq n}$ as a presentable idempotent: that is, the functor
\[ \operatorname{\mathcal{S}}^{\leq n} \rightarrow \operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{S}}^{\leq n} \quad \quad X \mapsto \Delta ^0 \otimes X \]
is an equivalence of $\infty $-categories. This follows from Example 9.7.6.4, since the $\infty $-category $\operatorname{\mathcal{S}}^{\leq n}$ is locally $n$-truncated.
Example 9.7.8.3. The $\infty $-category of spaces $\operatorname{\mathcal{S}}$ is a presentable idempotent. More precisely, the object $\Delta ^0 \in \operatorname{\mathcal{S}}$ exhibits $\operatorname{\mathcal{S}}$ as a presentable idempotent. This is a special case of Proposition 9.7.4.1.
Example 9.7.8.4. Let $\operatorname{\mathcal{S}}_{\ast }$ denote the $\infty $-category of pointed spaces (see Construction 5.5.2.1). Then $\operatorname{\mathcal{S}}_{\ast }$ is a presentable idempotent: see Proposition 
.
In what follows, we let $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ denote the $\infty $-category whose objects are presentable $\infty $-categories and whose morphisms are cocontinuous functors (Construction 9.5.3.1).
Definition 9.7.8.5. Let $\operatorname{\mathcal{E}}$ be a presentable $\infty $-category and let $E \in \operatorname{\mathcal{E}}$ be an object which exhibits $\operatorname{\mathcal{E}}$ as a presentable idempotent. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is a presentable $\operatorname{\mathcal{E}}$-module if it is presentable and the functor
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}\widehat{\otimes } \operatorname{\mathcal{C}}\quad \quad X \mapsto E \otimes X \]
is an equivalence of $\infty $-categories. We let $\operatorname{Mod}_{ \operatorname{\mathcal{E}}}( \operatorname{\mathcal{QC}}^{\operatorname{LPr}} )$ denote the full subcategory of $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ spanned by the presentable $\operatorname{\mathcal{E}}$-modules.
Example 9.7.8.7. Let $n$ be an integer, so that $\operatorname{\mathcal{S}}^{\leq n}$ is a presentable idempotent (Example 9.7.8.2). A presentable $\infty $-category $\operatorname{\mathcal{C}}$ is an $\operatorname{\mathcal{S}}^{\leq n}$-module if and only if it is locally $n$-truncated. See Example 9.7.6.4.
Example 9.7.8.8. Every presentable $\infty $-category is a presentable $\operatorname{\mathcal{S}}$-module. This is a restatement of Proposition 9.7.4.1.
Example 9.7.8.9. Let $\operatorname{\mathcal{E}}$ be a presentable idempotent. Then $\operatorname{\mathcal{E}}$ is a presentable $\operatorname{\mathcal{E}}$-module.
Proposition 9.7.8.10. Let $\operatorname{\mathcal{E}}$ be a presentable idempotent. Then the $\infty $-category of presentable $\operatorname{\mathcal{E}}$-modules $\operatorname{Mod}_{\operatorname{\mathcal{E}}}( \operatorname{\mathcal{QC}}^{\operatorname{LPr}} )$ is a reflective and coreflective subcategory of $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$. That is, the inclusion functor $\iota : \operatorname{Mod}_{\operatorname{\mathcal{E}}}( \operatorname{\mathcal{QC}}^{\operatorname{LPr}} ) \hookrightarrow \operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ admits both left and right adjoints, given by the constructions $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{E}}\widehat{\otimes } \operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}\mapsto \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}})$, respectively.
Example 9.7.8.11. Let $n$ be an integer and let $\operatorname{\mathcal{E}}= \operatorname{\mathcal{S}}^{\leq n}$ be the presentable idempotent of Example 9.7.8.7. If $\operatorname{\mathcal{C}}$ is a presentable $\infty $-category, then $\operatorname{\mathcal{E}}\widehat{\otimes } \operatorname{\mathcal{C}}$ and $\operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}})$ can be identified with the full subcategories of $\operatorname{\mathcal{C}}$ spanned by the $n$-truncated and $n$-cotruncated objects, respectively (Proposition 9.7.6.5 and Corollary 8.4.4.5). They are universal among locally $n$-truncated presentable $\infty $-categories equipped with a cocontinuous functor from and to $\operatorname{\mathcal{C}}$, respectively.
We will give the proof of Proposition 9.7.8.10 at the end of this section. First, let us formulate the preceding definitions in somewhat greater generality.
Definition 9.7.8.12. Let $\mathbb {K}$ be a collection of simplicial sets. We say that an $\infty $-category $\operatorname{\mathcal{E}}$ is a $\mathbb {K}$-cocomplete idempotent if it is $\mathbb {K}$-cocomplete and there exists an object $E \in \operatorname{\mathcal{E}}$ which satisfies the following condition:
- $(\ast )$
For every $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{C}}$, the restriction functor
\[ \operatorname{Fun}^{\mathbb {K}-\mathrm{Bil}}( \operatorname{\mathcal{E}}\times \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}) \quad \quad F(-,-) \mapsto F(E,-) \]
is an equivalence of $\infty $-categories.
In this case, we say that the object $E \in \operatorname{\mathcal{E}}$ exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete idempotent.
Example 9.7.8.15. Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ be the $\infty $-category of Notation 9.7.4.2. Then $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ is a $\mathbb {K}$-cocomplete idempotent. More precisely, the object $\Delta ^0 \in \operatorname{\mathcal{S}}_{< \mathbb {K} }$ exhibits $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ as a $\mathbb {K}$-cocomplete idempotent: see Corollary 9.7.4.12.
Proposition 9.7.8.16. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ be $\mathbb {K}$-cocomplete $\infty $-categories, and let $E \in \operatorname{\mathcal{E}}$ be an object which exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete idempotent. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to $\operatorname{\mathcal{E}}\otimes _{\mathbb {K} } \operatorname{\mathcal{C}}_0$, for some $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{C}}_0$.
- $(1')$
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to $\operatorname{Fun}^{ \mathbb {K} -\mathrm{cocont}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}_0 )$, for some $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{C}}_0$.
- $(2)$
The functor
\[ F_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}\quad \quad X \mapsto E \otimes X \]
is an equivalence of $\infty $-categories.
- $(2')$
Evaluation at $E$ induces an equivalence of $\infty $-categories $G_{\operatorname{\mathcal{C}}}: \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.
- $(3)$
The functor $F_{\operatorname{\mathcal{C}}}$ admits a left homotopy inverse $\operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.
- $(3')$
The functor $G_{\operatorname{\mathcal{C}}}$ admits a right homotopy inverse $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}})$.
- $(3'')$
There exists a $\mathbb {K}$-bilinear functor $T: \operatorname{\mathcal{E}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ such that $T|_{ \{ E\} \times \operatorname{\mathcal{C}}}$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$.
Proof.
The equivalence $(3') \Leftrightarrow (3'')$ is immediate and the equivalence $(3) \Leftrightarrow (3'')$ follows from the definition of the $\mathbb {K}$-cocomplete tensor product. We will show that $(1) \Leftrightarrow (2) \Leftrightarrow (3)$; the equivalences $(1') \Leftrightarrow (2') \Leftrightarrow (3')$ follow from a similar argument. The implication $(2) \Rightarrow (1)$ is immediate. To prove the converse, suppose that $\operatorname{\mathcal{C}}$ is equivalent to a $\mathbb {K}$-cocomplete tensor product $\operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}_0$. In this case, we can use the associativity constraints of Proposition 9.7.7.1 to identify $F_{\operatorname{\mathcal{C}}}$ with the tensor product
\[ \operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}_0 \xrightarrow { F_{\operatorname{\mathcal{E}}} \otimes _{\mathbb {K} } \operatorname{id}} ( \operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{E}}) \otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}_0. \]
It will therefore suffice to show that $F_{\operatorname{\mathcal{E}}}$ is an equivalence of $\infty $-categories, which is a reformulation of our assumption that $E$ exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete idempotent.
We now complete the proof by showing that $(3) \Rightarrow (2)$ (the converse follows immediately from the definitions). Note if condition $(3)$ is satisfied, then $\operatorname{\mathcal{C}}$ can be realized as a retract of the $\mathbb {K}$-cocomplete tensor product $\operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}$. Consequently, to prove that $F_{\operatorname{\mathcal{C}}}$ is an equivalence of $\infty $-categories, it will suffice to show that $F_{ \operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}}$ is an equivalence of $\infty $-categories, which is a special case of the implication $(1) \Rightarrow (2)$.
$\square$
Definition 9.7.8.17. Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{E}}$ be an $\infty $-category which is a $\mathbb {K}$-cocomplete idempotent. We say that a $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{C}}$ is a $\mathbb {K}$-cocomplete $\operatorname{\mathcal{E}}$-module if it satisfies the equivalent conditions of Proposition 9.7.8.16.
Proposition 9.7.8.20. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ be $\mathbb {K}$-cocomplete $\infty $-categories, and let $E \in \operatorname{\mathcal{E}}$ be an object which exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete idempotent, so that evaluation and tensor product with $E$ determine $\mathbb {K}$-cocontinuous functors
\[ \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}) \xrightarrow {U_{\operatorname{\mathcal{C}}}} \operatorname{\mathcal{C}}\xrightarrow { T_{\operatorname{\mathcal{C}}} } \operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}. \]
Then:
- $(1)$
For every $\mathbb {K}$-cocomplete $\operatorname{\mathcal{E}}$-module $\operatorname{\mathcal{D}}$, precomposition with $T_{\operatorname{\mathcal{C}}}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
- $(2)$
For every $\mathbb {K}$-cocomplete $\operatorname{\mathcal{E}}$-module $\operatorname{\mathcal{D}}$, postcomposition with $U_{\operatorname{\mathcal{C}}}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{D}}, \operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$.
Proof.
Invoking the universal property of the tensor product (and Remark 9.7.3.3), we can restate $(1)$ and $(2)$ as follows:
- $(1')$
For every $\mathbb {K}$-cocomplete $\operatorname{\mathcal{E}}$-module $\operatorname{\mathcal{D}}$, postcomposition with $U_{\operatorname{\mathcal{D}}}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
- $(2')$
For every $\mathbb {K}$-cocomplete $\operatorname{\mathcal{E}}$-module $\operatorname{\mathcal{D}}$, precomposition with $T_{\operatorname{\mathcal{D}}}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{E}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$.
Both assertions are clear, since the assumption that $\operatorname{\mathcal{D}}$ is a $\mathbb {K}$-cocomplete $\operatorname{\mathcal{E}}$-module guarantees that the functors $T_{\operatorname{\mathcal{D}}}$ and $U_{\operatorname{\mathcal{D}}}$ are equivalences of $\infty $-categories.
$\square$
Proof of Proposition 9.7.8.10.
Let $\operatorname{\mathcal{E}}$ be a presentable idempotent; we wish to show that the full subcategory $\operatorname{Mod}_{ \operatorname{\mathcal{E}}}( \operatorname{\mathcal{QC}}^{\operatorname{LPr}} ) \subseteq \operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ is both reflective and coreflective. Fix a presentable $\infty $-category $\operatorname{\mathcal{C}}$. Using Proposition 9.7.5.1 and Corollary 9.5.5.16, we see that $\operatorname{\mathcal{E}}\widehat{\otimes } \operatorname{\mathcal{C}}$ and $\operatorname{Fun}^{\operatorname{\mathrm{cocont}}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{C}})$ are also presentable. Let $E \in \operatorname{\mathcal{E}}$ be an object which exhibits $\operatorname{\mathcal{E}}$ as a presentable idempotent, so that evaluation and tensor product determine cocontinuous functors
\[ \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}) \xrightarrow {U_{\operatorname{\mathcal{C}}}} \operatorname{\mathcal{C}}\xrightarrow { T_{\operatorname{\mathcal{C}}} } \operatorname{\mathcal{E}}\widehat{\otimes } \operatorname{\mathcal{C}}. \]
To complete the proof, it will suffice to show that $T_{\operatorname{\mathcal{C}}}$ and $U_{\operatorname{\mathcal{C}}}$ exhibit $\operatorname{\mathcal{E}}\widehat{\otimes } \operatorname{\mathcal{C}}$ and $\operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}})$ as a $\operatorname{Mod}_{ \operatorname{\mathcal{E}}}( \operatorname{\mathcal{QC}}^{\operatorname{LPr}} )$-reflection and a $\operatorname{Mod}_{\operatorname{\mathcal{E}}}( \operatorname{\mathcal{QC}}^{\operatorname{LPr}} )$-coreflection of $\operatorname{\mathcal{C}}$, respectively. This is a special case of Proposition 9.7.8.20.
$\square$
Example 9.7.8.21. Let $\kappa $ be a regular cardinal and let $n$ be an integer. If $\kappa $ is uncountable, then the $\infty $-category $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa }$ is a $\kappa $-cocomplete idempotent. Moreover, a $\kappa $-cocomplete $\infty $-category $\operatorname{\mathcal{C}}$ is a $\kappa $-cocomplete $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa }$-module if and only if it is locally $n$-truncated. See Corollary 9.7.6.3. If $\kappa = \aleph _0$, then a similar assertion holds for the $\infty $-category $\operatorname{\mathcal{S}}^{\leq n}_{\mathrm{fin} }$ (Variant 9.7.6.10).
We close this section with a few observations about the dependence of Definitions 9.7.8.12 and 9.7.8.17 on the collection $\mathbb {K}$.
Proposition 9.7.8.27. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{E}}$ be a $\kappa $-cocomplete $\infty $-category, and let $E \in \operatorname{\mathcal{E}}$ be an object. If $E$ exhibits $\operatorname{\mathcal{E}}$ as a $\kappa $-cocomplete idempotent, then it also exhibits $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{E}})$ as a $\lambda $-cocomplete idempotent. The converse holds if $\operatorname{\mathcal{E}}$ is idempotent-complete.
Proof of Proposition 9.7.8.27.
The first assertion is a special case of Remark 9.7.8.26 (see Example 9.7.1.18). To prove the second assertion, assume that $E$ exhibits $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{E}})$ as a $\lambda $-cocomplete idempotent, and let $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\otimes _{\kappa } \operatorname{\mathcal{E}}$ be the functor given on objects by $F(X) = E \otimes X$. Using Proposition 9.7.5.3 and Example 9.7.1.18, we see that the functor $\operatorname{Ind}_{\kappa }^{\lambda }(F)$ is an equivalence of $\infty $-categories, so that $F$ is a Morita equivalence (Exercise 9.3.2.7). If $\operatorname{\mathcal{E}}$ is idempotent-complete, it follows that $F$ is an equivalence of $\infty $-categories (see Proposition 8.5.6.11).
$\square$
Proposition 9.7.8.29. Let $\mathbb {K}_0 \subseteq \mathbb {K}$ be collections of simplicial sets, let $\operatorname{\mathcal{E}}_0$ be a $\mathbb {K}_0$-cocomplete idempotent, and let $\operatorname{\mathcal{E}}$ be a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_0$ relative to $\mathbb {K}_0$ (so that $\operatorname{\mathcal{E}}$ is a $\mathbb {K}$-cocomplete idempotent). Let $\operatorname{\mathcal{C}}$ be a $\mathbb {K}$-cocomplete $\infty $-category. Then $\operatorname{\mathcal{C}}$ is a $\mathbb {K}$-cocomplete $\operatorname{\mathcal{E}}$-module if and only if it is a $\mathbb {K}_0$-cocomplete $\operatorname{\mathcal{E}}_0$-module.
Proof.
Choose a functor $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}$ which exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_0$ relative to $\mathbb {K}_0$ and an object $E \in \operatorname{\mathcal{E}}_0$ which exhibits $\operatorname{\mathcal{E}}_0$ as a $\mathbb {K}_0$-cocomplete idempotent. We then have a commutative diagram
\[ \xymatrix { \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont} }( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}) \ar [rr]^{ \circ F } \ar [dr] & & \operatorname{Fun}^{\mathbb {K}_0-\mathrm{cocont} }( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{C}}) \ar [dl] \\ & \operatorname{\mathcal{C}}& } \]
where the vertical maps are given by evaluation at the objects $F(E)$ and $E$, respectively. Our assumption on $F$ guarantees that the horizontal map is an equivalence of $\infty $-categories. It follows that the left vertical map is an equivalence if and only if the right vertical map is an equivalence.
$\square$