Proposition 9.7.5.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then the cocomplete tensor product $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$ is also presentable.
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9.7.5 Tensor Products of Presentable $\infty $-Categories
We now restrict the cocomplete tensor product of ยง9.7.3 to the setting of presentable $\infty $-categories.
The proof of Proposition 9.7.5.1 will require some preliminaries. We first study the dependence of the $\mathbb {K}$-cocomplete tensor product $\otimes _{\mathbb {K}}$ on the choice of $\mathbb {K}$.
Construction 9.7.5.2. Let $\mathbb {K} \subseteq \widehat{\mathbb {K}}$ be collections of simplicial sets. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\mathbb {K}$-cocomplete $\infty $-categories, let $\widehat{\operatorname{\mathcal{C}}}$ and $\widehat{\operatorname{\mathcal{D}}}$ be $\widehat{\mathbb {K}}$-cocomplete $\infty $-categories, and suppose we are given $\mathbb {K}$-cocontinuous functors $F: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$. Then $F$ and $G$ induce a $\mathbb {K}$-cocontinuous functor which is characterized (up to isomorphism) by the requirement that the diagram of $\infty $-categories commutes up to isomorphism.
\[ (F \otimes G): \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{C}}} \otimes _{ \widehat{\mathbb {K}} } \widehat{\operatorname{\mathcal{D}}} \]
\[ \xymatrix { \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\ar [r] \ar [d]^{F \times G} & \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}\ar [d]^{ F \otimes G} \\ \widehat{\operatorname{\mathcal{C}}} \times \widehat{\operatorname{\mathcal{D}}} \ar [r]^{T} & \widehat{\operatorname{\mathcal{C}}} \otimes _{ \widehat{\mathbb {K}} } \widehat{\operatorname{\mathcal{D}}} } \]
Proposition 9.7.5.3. In the situation of Construction 9.7.5.2, suppose that the functors $F$ and $G$ exhibit $\widehat{\operatorname{\mathcal{C}}}$ and $\widehat{\operatorname{\mathcal{D}}}$ as $\widehat{\mathbb {K}}$-cocompletions of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ relative to $\mathbb {K}$, respectively (see Definition 9.7.1.16). Then the functor $F \otimes G$ exhibits $\widehat{\operatorname{\mathcal{C}}} \otimes _{ \widehat{\mathbb {K}}} \widehat{\operatorname{\mathcal{D}}}$ as a $\widehat{\mathbb {K}}$-cocompletion of $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$ relative to $\mathbb {K}$.
Proof. Let $\widehat{\operatorname{\mathcal{E}}}$ be a $\widehat{\mathbb {K}}$-cocomplete $\infty $-category; we wish to show that precomposition with $F \otimes G$ induces an equivalence of $\infty $-categories
\[ \operatorname{Fun}^{ \widehat{\mathbb {K}} -\mathrm{cocont} }( \widehat{\operatorname{\mathcal{C}}} \otimes _{ \widehat{\mathbb {K} } } \widehat{\operatorname{\mathcal{D}}}, \widehat{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}^{ \mathbb {K} -\mathrm{cocont} }( \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}, \widehat{\operatorname{\mathcal{E}}} ). \]
Invoking the universal property of the cocomplete tensor product (and the construction of $F \otimes G$), we can identify this with the functor
\[ \operatorname{Fun}^{ \widehat{\mathbb {K}}-\mathrm{Bil}}( \widehat{\operatorname{\mathcal{C}}} \times \widehat{\operatorname{\mathcal{D}}}, \widehat{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}^{ \mathbb {K} -\mathrm{Bil}}( \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}, \widehat{\operatorname{\mathcal{E}}} ) \]
given by precomposition with $F \times G$. Using Remark 9.7.3.3, we can rewrite this functor as a composition
\begin{eqnarray*} \operatorname{Fun}^{ \widehat{\mathbb {K} } -\mathrm{cocont} }( \widehat{\operatorname{\mathcal{C}}}, \operatorname{Fun}^{ \widehat{\mathbb {K} } -\mathrm{cocont}}( \widehat{\operatorname{\mathcal{D}}}, \widehat{\operatorname{\mathcal{E}}} ) ) & \xrightarrow { \circ F} & \operatorname{Fun}^{ \mathbb {K} -\mathrm{cocont} }( \operatorname{\mathcal{C}}, \operatorname{Fun}^{ \widehat{\mathbb {K} } -\mathrm{cocont} }( \widehat{\operatorname{\mathcal{D}}}, \widehat{\operatorname{\mathcal{E}}} ) ) \\ & \xrightarrow { \circ G} & \operatorname{Fun}^{ \mathbb {K} -\mathrm{cocont} }( \operatorname{\mathcal{C}}, \operatorname{Fun}^{ \mathbb {K} -\mathrm{cocont} }( \operatorname{\mathcal{D}}, \widehat{\operatorname{\mathcal{E}}} ) ) \end{eqnarray*}
which is an equivalence by virtue of our assumptions on $F$ and $G$. $\square$
Example 9.7.5.4. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are $\kappa $-cocomplete. For every regular cardinal $\lambda $ satisfying $\kappa \trianglelefteq \lambda $, the $\infty $-categories $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ and $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ are $\lambda $-cocomplete (Corollary 9.5.5.6), and can be viewed as $\lambda $-cocompletions of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ relative to the collection of all $\kappa $-small simplicial sets (Example 9.7.1.18). In this case, Proposition 9.7.5.3 supplies an equivalence of $\infty $-categories
\[ \operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}}\otimes _{\kappa } \operatorname{\mathcal{D}}) \xrightarrow {\sim } \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \otimes _{\lambda } \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}}). \]
Corollary 9.7.5.5. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\lambda $-cocomplete $\infty $-categories. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $(\kappa ,\lambda )$-compactly generated, then the tensor product $\operatorname{\mathcal{C}}\otimes _{\lambda } \operatorname{\mathcal{D}}$ is also $(\kappa ,\lambda )$-compactly generated.
Proof. Combine Example 9.7.5.4 with Proposition 9.4.1.11. $\square$
Corollary 9.7.5.6. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be cocomplete $\infty $-categories. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\kappa $-compactly generated, then the cocomplete tensor product $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$ is $\kappa $-compactly generated.
Proof. Apply Corollary 9.7.5.5 in the special case where $\lambda = \Omega $ is a strongly inaccessible cardinal. $\square$
Corollary 9.7.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be cocomplete $\infty $-categories. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are compactly generated, then the cocomplete tensor product $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$ is compactly generated.
Proof. Apply Corollary 9.7.5.6 in the special case $\kappa = \aleph _0$. $\square$
We now prove a more quantitative version of Proposition 9.7.5.1.
Proposition 9.7.5.8. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are $\kappa $-presentable. Then the cocomplete tensor product $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$ is $\kappa $-presentable.
Proof. By virtue of Proposition 9.5.5.3, we may assume that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\operatorname{Ind}_{\kappa }$-completions of $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{D}}_0$ which are essentially small and $\kappa $-cocomplete. Example 9.7.5.4 then guarantees that $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$ can be realized as an $\operatorname{Ind}_{\kappa }$-completion of the $\kappa $-cocomplete tensor product $\operatorname{\mathcal{C}}_0 \otimes _{\kappa } \operatorname{\mathcal{D}}_0$, which is also essentially small (Remark 9.7.3.16). $\square$
Proof of Proposition 9.7.5.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. By virtue of Remark 9.5.5.2, there exists a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\kappa $-presentable. Applying Proposition 9.7.5.8, we conclude that $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$ is also $\kappa $-presentable; in particular, it is presentable. $\square$
It will sometimes be useful to have a more explicit description of the cocomplete tensor product $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$.
Proposition 9.7.5.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then there is a canonical equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}\simeq \operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{C}})$. Here $\operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{C}})$ denotes the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{C}})$ spanned by those functors which preserve small limits.
Remark 9.7.5.10. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are presentable $\infty $-categories, then a functor $G: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ preserves small limits if and only if it admits a left adjoint. This follows by applying Proposition 9.5.2.2 to the opposite functor $G^{\operatorname{op}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.
Proof of Proposition 9.7.5.9. Recall that if $\operatorname{\mathcal{E}}$ is a presentable $\infty $-category, then the covariant Yoneda embedding supplies an equivalence from $\operatorname{\mathcal{E}}$ to the full subcategory $\operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{E}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{E}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ spanned by those functors which preserve small limits (Theorem 9.5.1.8). Passing to opposite $\infty $-categories, we obtain an equivalence $\operatorname{\mathcal{E}}^{\operatorname{op}} \simeq \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}}^{\operatorname{op}} )$. Applying this observation for $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}$ (together with Remark 9.7.3.3), we obtain equivalences
\begin{eqnarray*} (\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}})^{\operatorname{op}} & \simeq & \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}^{\operatorname{op}} ) \\ & \simeq & \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{D}}, \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{\operatorname{op}} ) ) \\ & \simeq & \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}^{\operatorname{op}} ). \\ \end{eqnarray*}
Passing to opposite $\infty $-categories again, we obtain an equivalence of $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$ with $\operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{C}})$. $\square$
Example 9.7.5.11. Let $\operatorname{\mathcal{D}}$ be a presentable $\infty $-category. It follows from Proposition 9.7.4.1 that the functor is an equivalence of $\infty $-categories. Under the equivalence of Proposition 9.7.5.9, this corresponds to the covariant Yoneda embedding $\operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (which is an equivalence by Theorem 9.5.1.8).
\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}\widehat{\otimes } \operatorname{\mathcal{D}}\quad \quad X \mapsto \Delta ^0 \otimes X \]
Remark 9.7.5.12 (Functoriality). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ be cocontinuous functors between presentable $\infty $-categories, so that $F$ and $G$ determine a cocontinuous functor $(F \widehat{\otimes } G): \operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \widehat{\otimes } \operatorname{\mathcal{D}}'$. Under the equivalences of Proposition 9.7.5.9, $F \widehat{\otimes } G$ corresponds to a cocontinuous functor from $\operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{C}})$ to $\operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}'^{\operatorname{op}}, \operatorname{\mathcal{C}}' )$. Unwinding the definitions, we see that this is left adjoint to the functor where $F^{R}$ denotes a right adjoint to the functor $F$.
\[ \operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}'^{\operatorname{op}}, \operatorname{\mathcal{C}}' ) \rightarrow \operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{C}}) \quad \quad T \mapsto F^{R} \circ T \circ G^{\operatorname{op}}, \]
Proposition 9.7.5.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ be cocontinuous functors of presentable $\infty $-categories. If $F$ and $G$ are Bousfield localization functors, then the cocomplete tensor product $(F \widehat{\otimes } G): \operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \widehat{\otimes } \operatorname{\mathcal{D}}'$ is a Bousfield localization functor.
Proof. Since the functor $F \widehat{\otimes } G$ factors as a composition of $F \widehat{\otimes } \operatorname{id}_{\operatorname{\mathcal{D}}}$ with $\operatorname{id}_{\operatorname{\mathcal{C}}'} \widehat{\otimes } G$, we may assume without loss of generality that either $F$ or $G$ is an identity functor. By symmetry, it suffices to treat the case where $G = \operatorname{id}_{\operatorname{\mathcal{D}}}$. Our assumption that $F$ is a Bousfield localization functor guarantees that it admits a fully faithful right adjoint $F^{R}: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$. Invoking Remark 9.7.5.12, we see that the right adjoint to $F \widehat{\otimes } \operatorname{id}_{\operatorname{\mathcal{D}}}$ can be identified with the functor $\operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{C}}' ) \rightarrow \operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{C}})$ given by postcomposition with $F^{R}$, which is also fully faithful. $\square$