Subsection 9.7.3 (0707): Tensor Products of $\infty $-Categories

9.7.3 Tensor Products of $\infty $-Categories

Let $\mathbb {K}$ be a collection of simplicial sets. Recall that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocontinuous if it preserves $K$-indexed colimits, for each $K \in \mathbb {K}$. This condition has a counterpart for functors of two variables.

Definition 9.7.3.1. Let $\mathbb {K}$ be a collection of simplicial sets and suppose we are given $\infty $-categories $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$. We say that a functor $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is $\mathbb {K}$-bilinear if it satisfies the following conditions:

For every object $C \in \operatorname{\mathcal{C}}$, the functor

\[ T(C, \bullet ): \operatorname{\mathcal{D}}\simeq \{ C \} \times \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\xrightarrow {T} \operatorname{\mathcal{E}} \]

is $\mathbb {K}$-cocontinuous.
For every object $D \in \operatorname{\mathcal{D}}$, the functor

\[ T(\bullet , D): \operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}\times \{ D \} \hookrightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\xrightarrow {T} \operatorname{\mathcal{E}} \]

is $\mathbb {K}$-cocontinuous.

We let $\operatorname{Fun}^{ \mathbb {K}-\mathrm{Bil} }( \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ spanned by the $\mathbb {K}$-bilinear functors. We will be particularly interested in the special case where $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets for some regular cardinal $\kappa $. In this case, we say that $T$ is $\kappa $-bilinear if it is $\mathbb {K}$-bilinear. We say that $T$ is bilinear if it is $\aleph _0$-bilinear: that is, it preserves finite colimits in each variable.

Warning 9.7.3.2. In the situation of Definition 9.7.3.1, the condition that the functor $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is $\mathbb {K}$-bilinear depends not only on $T$ as an abstract functor, but on the decomposition of the domain of $T$ as a product of $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.

Remark 9.7.3.3. Let $\mathbb {K}$ be a collection of simplicial sets, and let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be $\mathbb {K}$-cocomplete $\infty $-categories. Then the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ is also $\mathbb {K}$-cocomplete, and $K$-indexed colimits in $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ are computed levelwise for each $K \in \mathbb {K}$ (Proposition 7.1.8.2). Moreover, the full subcategory $\operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ of $\mathbb {K}$-cocontinuous functors is closed under $K$-indexed colimits for $K \in \mathbb {K}$, and is therefore also $\mathbb {K}$-cocomplete (Remark 7.6.6.23). It follows that the canonical isomorphism of simplicial sets $\operatorname{Fun}( \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) )$ restricts to an isomorphism of $\infty $-categories

\[ \operatorname{Fun}^{ \mathbb {K} -\mathrm{Bil} }( \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) ). \]

Similarly, we can identify $\operatorname{Fun}^{ \mathbb {K} -\mathrm{Bil} }( \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ with $\operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{D}}, \operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) )$.

Example 9.7.3.4. Let $\kappa $ be an uncountable regular cardinal. Using Remark 4.9.5.10 and Example 7.6.1.19, we see that the $\infty $-category $\operatorname{\mathcal{S}}_{< \kappa }$ admits finite products, so that the formation of cartesian products defines a functor of $\infty $-categories

\[ \operatorname{\mathcal{S}}_{< \kappa } \times \operatorname{\mathcal{S}}_{< \kappa } \rightarrow \operatorname{\mathcal{S}}_{< \kappa } \quad \quad (X,Y) \mapsto X \times Y. \]

This functor is $\kappa $-bilinear: this follows from the universality of $\kappa $-small colimits in $\operatorname{\mathcal{S}}_{< \kappa }$ (see Proposition 7.7.6.7).

Proposition 9.7.3.5. Let $\mathbb {K}$ be a collection of simplicial sets. Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be $\mathbb {K}$-cocomplete $\infty $-categories, and let $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor. Then:

$(1)$: If $T$ is $\mathbb {K}$-cocontinuous and each $K \in \mathbb {K}$ is weakly contractible, then $T$ is $\mathbb {K}$-bilinear.
$(2)$: If $T$ is $\mathbb {K}$-bilinear and each $K \in \mathbb {K}$ is sifted, then $T$ is $\mathbb {K}$-cocontinuous.

Proof. Assertion $(1)$ follows from the observation that if a simplicial set $K \in \mathbb {K}$ is weakly contractible, then every constant map $K^{\triangleright } \rightarrow \{ C\} \hookrightarrow \operatorname{\mathcal{C}}$ or $K^{\triangleright } \rightarrow \{ D\} \hookrightarrow \operatorname{\mathcal{D}}$ is a colimit diagram (Corollary 7.2.3.5). To prove $(2)$, assume that $T$ is $\mathbb {K}$-bilinear and suppose we are given a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$, where $K \in \mathbb {K}$ is sifted; we wish to show that $T \circ \overline{q}$ is a colimit diagram in $\operatorname{\mathcal{E}}$. Note that $\overline{q}$ can be identified with a pair $( \overline{q}_{\operatorname{\mathcal{C}}}, \overline{q}_{ \operatorname{\mathcal{D}}} )$, where $\overline{q}_{\operatorname{\mathcal{C}}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ and $\overline{q}_{\operatorname{\mathcal{D}}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ are colimit diagrams in $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively (Example 7.1.3.11). Let $F$ denote the composition

\[ K^{\triangleright } \times K^{\triangleright } \xrightarrow { \overline{q}_{\operatorname{\mathcal{C}}} \times \overline{q}_{\operatorname{\mathcal{D}}} } \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\xrightarrow {T} \operatorname{\mathcal{E}} \]

and let $F_0$ denote the restriction of $F$ to $(K \times K)^{\triangleright }$. Our assumption that $T$ is $\mathbb {K}$-bilinear guarantees that $F_0$ is a colimit diagram in $\operatorname{\mathcal{E}}$ (see Variant 7.3.8.5). Moreover, $T \circ \overline{q}$ can be identified with the composition

\[ K^{\triangleright } \xrightarrow { \delta ^{\triangleright } } (K \times K)^{\triangleright } \xrightarrow {F_0} \operatorname{\mathcal{E}}, \]

where $\delta : K \hookrightarrow K \times K$ is the diagonal map. Since $K$ is sifted, the map $\delta $ is right cofinal, so that $T \circ \overline{q}$ is also a colimit diagram in $\operatorname{\mathcal{E}}$ (Corollary 7.2.2.3). $\square$

Corollary 9.7.3.6. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be $\kappa $-cocomplete $\infty $-categories. Then a functor $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is $\kappa $-bilinear if and only if it is bilinear and preserves $\kappa $-small filtered colimits.

Proof. Let $\mathbb {K}$ be the collection of all $\kappa $-small filtered $\infty $-categories. It follows from Corollary 9.2.2.23 that a functor $T$ is $\kappa $-bilinear if and only if it is both bilinear and $\mathbb {K}$-bilinear. The desired result now follows from Proposition 9.7.3.5, since every filtered $\infty $-category is sifted (Example 9.7.2.4). $\square$

Definition 9.7.3.7. Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are $\mathbb {K}$-cocomplete. We will say that a functor of $\infty $-categories $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ if the following conditions are satisfied:

$(1)$: The $\infty $-category $\operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete.
$(2)$: The functor $T$ is $\mathbb {K}$-bilinear.
$(3)$: For every $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{E}}'$, precomposition with $T$ induces an equivalence of $\infty $-categories

\[ \operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ) \rightarrow \operatorname{Fun}^{ \mathbb {K} -\mathrm{Bil}}( \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}' ). \]

Remark 9.7.3.8. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\mathbb {K}$-cocomplete $\infty $-categories, and let $Q$ denote the collection of all diagrams

\[ \overline{q} = ( \overline{q}_{\operatorname{\mathcal{C}}}, \overline{q}_{\operatorname{\mathcal{D}}} ): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}} \]

where $K \in \mathbb {K}$ and one of the following conditions is satisfied:

The diagram $\overline{q}_{\operatorname{\mathcal{C}}}$ is constant and $\overline{q}_{\operatorname{\mathcal{D}}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.
The diagram $\overline{q}_{\operatorname{\mathcal{D}}}$ is constant and $\overline{q}_{\operatorname{\mathcal{C}}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Then a functor $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ (in the sense of Definition 9.7.3.7) if and only if it exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ relative to $Q$ (in the sense of Definition 9.7.1.1).

Proposition 9.7.3.9 (Existence of Cocomplete Tensor Products). Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\mathbb {K}$-cocomplete $\infty $-categories. Then there exists a functor of $\infty $-categories $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ which exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$.

Proof. By virtue of Remark 9.7.3.8, this is a special case of Theorem 9.7.1.4. $\square$

Notation 9.7.3.10. Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\mathbb {K}$-cocomplete $\infty $-categories. Proposition 9.7.3.9 guarantees that there exists a functor $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ which exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$. It follows immediately from the definitions that the $\infty $-category $\operatorname{\mathcal{E}}$ (and the functor $T$) are uniquely determined up to equivalence. To emphasize this uniqueness, we will typically denote the $\infty $-category $\operatorname{\mathcal{E}}$ by $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$, and refer to it as the $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$. For objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, we will sometimes write $C \otimes D$ for the object $T(C,D) \in \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$.

Remark 9.7.3.11. In the situation of Notation 9.7.3.10, let $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$ be the smallest replete full subcategory of $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$ which contains every object of the form $C \otimes D$ and is closed under $K$-indexed colimits for $K \in \mathbb {K}$. Invoking the universal property of the tensor product, we deduce that there is an (essentially unique) $\mathbb {K}$-cocontinuous functor $F: \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ such that $(F \circ T): \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is isomorphic to $T$. Invoking the universal property again, we see that the composition

\[ \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}\xrightarrow {F} \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}} \]

is isomorphic to the identity functor. It follows that $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$. Stated more informally, the $\infty $-category $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$ is generated by objects of the form $C \otimes D$ under $K$-indexed colimits for $K \in \mathbb {K}$ (this also follows from the proof of Theorem 9.7.1.4 given in §9.7.1).

Example 9.7.3.12. Let $\mathbb {K}$ be a collection of sifted simplicial sets. Then, for every pair of $\mathbb {K}$-cocomplete $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, the $\mathbb {K}$-cocomplete tensor product $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$ can be identified with the cartesian product $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$. More precisely, a functor $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ if and only if it is an equivalence of $\infty $-categories. This follows from the characterization of $\mathbb {K}$-bilinear functors supplied by Proposition 9.7.3.5.

Remark 9.7.3.13 (Symmetry). Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\mathbb {K}$-cocomplete $\infty $-categories. Then a functor $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ if and only if it exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{D}}$ with $\operatorname{\mathcal{C}}$, when regarded as a functor from $\operatorname{\mathcal{D}}\times \operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{E}}$. Stated more informally, we have a canonical equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{D}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}$.

Variant 9.7.3.14. Let $\kappa $ be a regular cardinal, and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\kappa $-cocomplete $\infty $-categories. We say that a functor $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ exhibits $\operatorname{\mathcal{E}}$ as a $\kappa $-cocomplete tensor product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ if it exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$, where $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets. In this case, we will typically denote the $\infty $-category $\operatorname{\mathcal{E}}$ by $\operatorname{\mathcal{C}}\otimes _{\kappa } \operatorname{\mathcal{D}}$.

Variant 9.7.3.15. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be cocomplete $\infty $-categories. We say that a functor $T: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ exhibits $\operatorname{\mathcal{E}}$ as a cocomplete tensor product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ if it exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$, where $\mathbb {K}$ is the collection of all small simplicial sets. In this case, we will typically denote the $\infty $-category $\operatorname{\mathcal{E}}$ by $\operatorname{\mathcal{C}}\widehat{\otimes } \operatorname{\mathcal{D}}$. Note that this can be regarded as a special case of Variant 9.7.3.14 (taking $\kappa = \Omega $ to be the strongly inaccessible cardinal of Remark 4.9.0.4).

Remark 9.7.3.16 (Size Estimates for Tensor Products). Let $\kappa < \lambda $ be regular cardinals, let $\mathbb {K}$ be a collection of $\kappa $-small simplicial sets, and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\mathbb {K}$-cocomplete $\infty $-categories. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\lambda $-small and $\lambda $ has exponential cofinality $\geq \kappa $, then the $\mathbb {K}$-cocomplete tensor product $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$ is also essentially $\lambda $-small. By virtue of Remark 9.7.3.8, this is a special case of Corollary 9.7.1.15.

In particular, if $\mathbb {K}$ is a small collection of small simplicial sets and the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially small, then $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K}} \operatorname{\mathcal{D}}$ is also essentially small.