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9.7.7 Associativity

Let $\mathbb {K}$ be a collection of simplicial sets, which we regard as fixed throughout this section. In ยง9.7.3, we showed that every pair of $\mathbb {K}$-cocomplete $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ admit a $\mathbb {K}$-cocomplete tensor product $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}$, which is universal among $\infty $-categories equipped with a $\mathbb {K}$-bilinear functor

\[ T_{\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}}: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}\quad \quad (C,D) \mapsto C \otimes D \]

(Definition 9.7.3.7). The definition is manifestly symmetric in $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, so we have a canonical equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\otimes _{\mathbb {K}} \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{D}}\otimes _{\mathbb {K}} \operatorname{\mathcal{C}}$ (Remark 9.7.3.13). Stated more informally, the cocomplete tensor product $\widehat{\otimes }$ is commutative. Our goal in this section is to show that it is also associative. More precisely, we have the following:

Proposition 9.7.7.1 (Associativity Constraints). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be $\mathbb {K}$-cocomplete $\infty $-categories. Then there is an equivalence of $\infty $-categories

\[ \alpha _{ \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}}: \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } ( \operatorname{\mathcal{D}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{E}}) \rightarrow (\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}) \otimes _{ \mathbb {K} } \operatorname{\mathcal{E}} \]

which is characterized (up to isomorphism) by the requirement that the diagram

\[ \xymatrix { & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}\ar [dl] \ar [dr] & \\ \operatorname{\mathcal{C}}\otimes _{\mathbb {K}} (\operatorname{\mathcal{D}}\otimes _{\mathbb {K} } \operatorname{\mathcal{E}}) \ar [rr]^{\alpha _{\operatorname{\mathcal{C}},\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}} } & & (\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}) \otimes _{ \mathbb {K} } \operatorname{\mathcal{E}}) } \]

commutes (up to isomorphism). Here the left and right vertical maps are the functors given on objects by $(C,D,E) \mapsto C \otimes (D \otimes E)$ and $(C,D,E) \mapsto (C \otimes D) \otimes E$, respectively.

To prove Proposition 9.7.7.1, it will be convenient to work with a generalization of Definition 9.7.3.1.

Definition 9.7.7.2. Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a finite collection of $\mathbb {K}$-cocomplete $\infty $-categories. We say that a functor of $\infty $-categories $T: \prod _{i \in I} \operatorname{\mathcal{C}}_{i} \rightarrow \operatorname{\mathcal{D}}$ is $\mathbb {K}$-multilinear if, for every $j \in I$ and every collection of objects $\{ C_ k \in \operatorname{\mathcal{C}}_{k} \} _{k \neq j}$, the functor

\[ \operatorname{\mathcal{C}}_ j \simeq \operatorname{\mathcal{C}}_{j} \times \prod _{k \neq j} \{ C_ k \} \hookrightarrow \prod _{i \in I} \operatorname{\mathcal{C}}_ i \xrightarrow {T} \operatorname{\mathcal{D}} \]

is $\mathbb {K}$-cocontinuous. We let $\operatorname{Fun}^{ \mathbb {K}-\mathrm{Mult} }( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{D}})$ spanned by the $\mathbb {K}$-multilinear functors from $\prod _{i \in I} \operatorname{\mathcal{C}}_ i$ to $\operatorname{\mathcal{D}}$.

Example 9.7.7.3. Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a finite collection of $\mathbb {K}$-cocomplete $\infty $-categories and suppose we are given a functor $T: \prod _{i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}$. Then:

  • If $I = \emptyset $, then $T$ is automatically $\mathbb {K}$-multilinear.

  • If $I = \{ i\} $ consists of a single element, then $T: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}$ is $\mathbb {K}$-multilinear if and only if it is $\mathbb {K}$-cocontinuous.

  • If $I = \{ i,j\} $ consists of two elements, then $T: \operatorname{\mathcal{C}}_ i \times \operatorname{\mathcal{C}}_ j \rightarrow \operatorname{\mathcal{D}}$ is $\mathbb {K}$-multilinear if and only if it is $\mathbb {K}$-bilinear, in the sense of Definition 9.7.3.1.

Warning 9.7.7.4. In the situation of Definition 9.7.7.2, the condition that a functor $T: \prod _{i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}$ is $\mathbb {K}$-multilinear depends not only on the functor $T$, but also on the decomposition of its domain as a product (see Warning 9.7.3.2).

Remark 9.7.7.5. In the situation of Definition 9.7.7.2, suppose that $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete. Then, for each $K \in \mathbb {K}$, the full subcategory $\operatorname{Fun}^{ \mathbb {K}-\mathrm{Mult} }( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{D}})$ is closed under (levelwise) $K$-indexed colimits; in particular, $\operatorname{Fun}^{ \mathbb {K}-\mathrm{Mult} }( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{D}})$ is also $\mathbb {K}$-cocomplete (Remark 7.6.6.23). Moreover, for every subset $J \subseteq I$, the canonical isomorphism $\operatorname{Fun}( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}(\prod _{j \in J} \operatorname{\mathcal{C}}_ j, \operatorname{Fun}( \prod _{ k \in I \setminus J} \operatorname{\mathcal{C}}_ k, \operatorname{\mathcal{D}})$ restricts to an isomorphism of full subcategories

\[ \operatorname{Fun}^{\mathbb {K}-\mathrm{Mult}}( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{\mathbb {K}-\mathrm{Mult}}( \prod _{j \in J} \operatorname{\mathcal{C}}_ j, \operatorname{Fun}^{\mathbb {K}-\mathrm{Mult}}( \prod _{ k \in I \setminus J} \operatorname{\mathcal{C}}_ k, \operatorname{\mathcal{D}})). \]

Compare with Remark 9.7.3.3.

Definition 9.7.7.6. Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a finite collection of $\mathbb {K}$-cocomplete $\infty $-categories. We say that a functor $T: \prod _{i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocomplete tensor product of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ if the following conditions are satisfied:

$(1)$

The $\infty $-category $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete.

$(2)$

The functor $T$ is $\mathbb {K}$-multilinear.

$(3)$

For every $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $T$ induces an equivalence of $\infty $-categories

\[ \operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}^{ \mathbb {K} -\mathrm{Mult}}( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{E}}). \]

Example 9.7.7.7 (Few Factors). Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ and $T: \prod _{i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}$ be as in Definition 9.7.7.6. Then:

  • If $I = \{ i, j\} $ has two elements, then $T$ exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocomplete tensor product of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ (in the sense of Definition 9.7.7.6) if and only if it exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{C}}_ i$ with $\operatorname{\mathcal{C}}_ j$ (in the sense of Definition 9.7.3.7).

  • If $I = \{ i\} $ has one element, then $T$ exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocomplete tensor product of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ if and only if it is an equivalence of $\infty $-categories.

  • If $I$ is empty, then $T$ exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocomplete tensor product of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ if and only if it exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocompletion of $\Delta ^{0} \simeq \prod _{i \in I} \operatorname{\mathcal{C}}_ i$, in the sense of Definition 8.4.6.1. In this case, we can identify $\operatorname{\mathcal{D}}$ with the $\infty $-category $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ of Notation 9.7.4.2 (see Remark 9.7.4.7).

Remark 9.7.7.8. It follows immediately from the definitions that if a finite collection of $\mathbb {K}$-cocomplete $\infty $-categories $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ admits a $\mathbb {K}$-cocomplete tensor product, then that tensor product is uniquely determined up to equivalence. More precisely, suppose we are given $\mathbb {K}$-multilinear functors

\[ \operatorname{\mathcal{D}}\xleftarrow {T} \prod _{i \in I} \operatorname{\mathcal{C}}_ i \xrightarrow {T'} \operatorname{\mathcal{D}}', \]

where $T$ exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocomplete tensor product of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I }$. It follows from the universal property of Definition 9.7.7.6 that there exists a $\mathbb {K}$-cocontinuous functor $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ such that $U \circ T$ is isomorphic to $T'$ (as an object of $\operatorname{Fun}( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{D}}' )$); moreover, $U$ is uniquely determined up to isomorphism. In this case, $T'$ exhibits $\operatorname{\mathcal{D}}'$ as a $\mathbb {K}$-cocomplete tensor product of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I }$ if and only if $U$ is an equivalence of $\infty $-categories.

For existence, we have the following:

Proposition 9.7.7.9 (Existence of Cocomplete Tensor Products). Let $\{ \operatorname{\mathcal{C}}_ i \} _{ i \in I}$ be a finite collection of $\mathbb {K}$-cocomplete $\infty $-categories. Then there exists a functor of $\infty $-categories $T: \prod _{ i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocomplete tensor product of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$.

Proof. As with Proposition 9.7.3.9, this can be regarded as a special case of Theorem 9.7.1.4. Alternatively, if the set $I$ is nonempty, then the existence of $I$-indexed tensor products can be reduced to the existence of pairwise tensor products by repeated application of Corollary 9.7.7.11 below (the case $I = \emptyset $ follows from Example 9.7.7.7). $\square$

We can now state our main result:

Proposition 9.7.7.10 (Generalized Associativity). Let $\{ \operatorname{\mathcal{C}}_ i \} _{ i \in I }$ and $\{ \operatorname{\mathcal{D}}_ j \} _{j \in J}$ be finite collections of $\mathbb {K}$-cocomplete $\infty $-categories. Suppose we are given a function $f: I \rightarrow J$ and, for each $j \in J$, a functor $T_{j}: \prod _{ f(i) = j} \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}_ j $ which exhibits $\operatorname{\mathcal{D}}_ j$ as a tensor product of the collection $\{ \operatorname{\mathcal{C}}_ i \} _{f(i) = j}$. Then, for every $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $\prod _{j \in J} T_ j$ induces an equivalence of $\infty $-categories

\[ \theta : \operatorname{Fun}^{ \mathbb {K} -\mathrm{Mult} }( \prod _{j \in J} \operatorname{\mathcal{D}}_ j, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}^{ \mathbb {K}-\mathrm{Mult} }( \prod _{i \in I} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{E}}). \]

Proof. We proceed by induction on the cardinality of $J$. If $J$ is empty, then $\theta $ identifies with the identity functor $\operatorname{id}_{\operatorname{\mathcal{E}}}$ (Example 9.7.7.3) and there is nothing to prove. If $J = \{ j\} $ has a single element, then the desired result follows immediately from the hypothesis on $T_{j}$. Let us therefore assume that $J$ has at least two elements, so that there is a nonempty proper subset $J' \subset J$. Set $I' = f^{-1}(J')$. Using Remark 9.7.7.5, we can identify $\theta $ with the composition

\[ \xymatrix { \operatorname{Fun}^{ \mathbb {K} -\mathrm{Mult} }( \prod _{j \in J'} \operatorname{\mathcal{D}}_ j, \operatorname{Fun}^{\mathbb {K}-\mathrm{Mult}}( \prod _{j \in J \setminus J'} \operatorname{\mathcal{D}}_ j, \operatorname{\mathcal{E}}) ) \ar [d]^{\theta '} \\ \operatorname{Fun}^{ \mathbb {K} -\mathrm{Mult} }( \prod _{i \in I'} \operatorname{\mathcal{C}}_ i, \operatorname{Fun}^{\mathbb {K}-\mathrm{Mult}}( \prod _{j \in J \setminus J'} \operatorname{\mathcal{D}}_ j, \operatorname{\mathcal{E}}) ) \ar [d]^{\theta ''} \\ \operatorname{Fun}^{ \mathbb {K} -\mathrm{Mult} }( \prod _{i \in I'} \operatorname{\mathcal{C}}_ i, \operatorname{Fun}^{\mathbb {K}-\mathrm{Mult}}( \prod _{i \in I \setminus I'} \operatorname{\mathcal{C}}_ i, \operatorname{\mathcal{E}}) ),} \]

where $\theta '$ and $\theta ''$ are equivalences of $\infty $-categories by virtue of our inductive hypothesis. $\square$

Corollary 9.7.7.11. Let $\{ \operatorname{\mathcal{C}}_ i \} _{ i \in I }$ and $\{ \operatorname{\mathcal{D}}_ j \} _{j \in J}$ be finite collections of $\mathbb {K}$-cocomplete $\infty $-categories. Suppose we are given a function $f: I \rightarrow J$ and, for each $j \in J$, a functor $T_{j}: \prod _{ f(i) = j} \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}_ j $ which exhibits $\operatorname{\mathcal{D}}_ j$ as a tensor product of the collection $\{ \operatorname{\mathcal{C}}_ i \} _{f(i) = j}$. Let $S: \prod _{j \in J} \operatorname{\mathcal{D}}_ j \rightarrow \operatorname{\mathcal{E}}$ be a functor which exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\{ \operatorname{\mathcal{D}}_ j \} _{j \in J}$. Then the composite functor

\[ \prod _{i \in I} \operatorname{\mathcal{C}}_ i \xrightarrow { \prod _{j \in J} T_ j } \prod _{j \in J} \operatorname{\mathcal{D}}_ j \xrightarrow {S} \operatorname{\mathcal{E}} \]

exhibits $\operatorname{\mathcal{E}}$ as a $\mathbb {K}$-cocomplete tensor product of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$.

Proof of Proposition 9.7.7.1. Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be $\mathbb {K}$-cocomplete $\infty $-categories, and let

\[ \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } ( \operatorname{\mathcal{D}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{E}}) \xleftarrow {T} \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}\xrightarrow { T' } ( \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}) \otimes _{ \mathbb {K} } \operatorname{\mathcal{E}} \]

denote the functors given on by objects by the formulae $T(C,D,E) = C \otimes (D \otimes E)$ and $T'(C,D,E) = (C \otimes D) \otimes E$. It follows from Corollary 9.7.7.11 that $T$ and $T'$ exhibit $\operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } ( \operatorname{\mathcal{D}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{E}})$ and $( \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}) \otimes _{ \mathbb {K} } \operatorname{\mathcal{E}}$ as $\mathbb {K}$-cocomplete tensor products of the finite collection $\{ \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}\} $, in the sense of Definition 9.7.7.6. Invoking Remark 9.7.7.8, we conclude that there is an (essentially unique) equivalence of $\infty $-categories

\[ \alpha _{\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}}: \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } (\operatorname{\mathcal{D}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{E}}\rightarrow ( \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{D}}) \otimes _{ \mathbb {K} } \operatorname{\mathcal{E}} \]

for which $T'$ is isomorphic to $\alpha _{\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}} \circ T$. $\square$

Remark 9.7.7.13 (The Pentagon Identity). To simplify notation, let us write $\otimes $ for the $\mathbb {K}$-cocomplete tensor product $\otimes _{\mathbb {K}}$. If $\operatorname{\mathcal{B}}$, $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are $\mathbb {K}$-cocomplete $\infty $-categories, then the diagram of equivalences

\[ \xymatrix@C =-30pt@R=30pt{ & \operatorname{\mathcal{B}}\otimes ((\operatorname{\mathcal{C}}\otimes \operatorname{\mathcal{D}}) \otimes \operatorname{\mathcal{E}}) \ar [rr]^-{ \alpha _{\operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}\otimes \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}} } & & (\operatorname{\mathcal{B}}\otimes (\operatorname{\mathcal{C}}\otimes \operatorname{\mathcal{D}})) \otimes \operatorname{\mathcal{E}}\ar [dr]^{ \alpha _{\operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}} \otimes \operatorname{id}} & \\ \operatorname{\mathcal{B}}\otimes (\operatorname{\mathcal{C}}\otimes (\operatorname{\mathcal{D}}\otimes \operatorname{\mathcal{E}}) ) \ar [ur]^{ \operatorname{id}\otimes \alpha _{\operatorname{\mathcal{C}},\operatorname{\mathcal{D}},\operatorname{\mathcal{E}}} } \ar [drr]_{ \alpha _{\operatorname{\mathcal{B}},\operatorname{\mathcal{C}},\operatorname{\mathcal{D}}\otimes \operatorname{\mathcal{E}}}} & & & & ((\operatorname{\mathcal{B}}\otimes \operatorname{\mathcal{C}}) \otimes \operatorname{\mathcal{D}}) \otimes \operatorname{\mathcal{E}}\\ & & (\operatorname{\mathcal{B}}\otimes \operatorname{\mathcal{C}}) \otimes (\operatorname{\mathcal{D}}\otimes \operatorname{\mathcal{E}}) \ar [urr]_{\alpha _{ \operatorname{\mathcal{B}}\otimes \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}} } & & } \]

commutes up to isomorphism. To prove this, let $F$ and $G$ denote the equivalences given by clockwise and counterclockwise composition around the diagram, and consider the functor

\[ T: \operatorname{\mathcal{B}}\times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{B}}\otimes (\operatorname{\mathcal{C}}\otimes (\operatorname{\mathcal{D}}\otimes \operatorname{\mathcal{E}}) ) \quad \quad (B,C,D,E) \mapsto B \otimes (C \otimes (D \otimes E) ). \]

Applying Corollary 9.7.7.11 twice, we see that $T$ exhibits $\operatorname{\mathcal{B}}\otimes (\operatorname{\mathcal{C}}\otimes (\operatorname{\mathcal{D}}\otimes \operatorname{\mathcal{E}}) )$ as a tensor product of the collection $\{ \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}\} $. Consequently, to show that the $\mathbb {K}$-cocontinuous functors $F$ and $G$ are isomorphic, it will suffice to show that the $\mathbb {K}$-multilinear functors

\[ (F \circ T), (G \circ T): \operatorname{\mathcal{B}}\times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}\rightarrow ( (\operatorname{\mathcal{B}}\otimes \operatorname{\mathcal{C}}) \otimes \operatorname{\mathcal{D}}) \otimes \operatorname{\mathcal{E}} \]

are isomorphic. This is clear, since both are isomorphic to the functor $(B,C,D,E) \mapsto ((B \otimes C) \otimes D) \otimes E$.

For every uncountable regular cardinal $\lambda $, let $\operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }$ denote the $\infty $-category whose objects are $\lambda $-small $\mathbb {K}$-cocomplete $\infty $-categories, and whose morphisms are $\mathbb {K}$-cocontinuous functors (Notation 7.6.6.26).

Proposition 9.7.7.14. Let $\kappa < \lambda $ be regular cardinals such that $\lambda $ has exponential cofinality $\geq \kappa $ and each $K \in \mathbb {K}$ is essentially $\kappa $-small. Then the $\mathbb {K}$-cocomplete tensor product determines a monoidal structure on the homotopy category $\operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }}$ (with associativity constraints given by Proposition 9.7.7.1 and unit given by the equivalence $\operatorname{\mathcal{S}}_{< \mathbb {K} } \otimes _{ \mathbb {K} } \operatorname{\mathcal{S}}_{< \mathbb {K} } \simeq \operatorname{\mathcal{S}}_{< \mathbb {K} }$ of Example 9.7.4.9).

Proof. The well-definedness of the functor

\[ \otimes _{ \mathbb {K} }: \operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }} \times \operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }} \rightarrow \operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }} \]

follows from Remark 9.7.3.16 (where the values on morphisms are given by Construction 9.7.5.2). Using Remark 9.7.7.13, we see that this operation supplies a nonunital monoidal structure on $\operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }}$ (with associativity constraints given by Proposition 9.7.7.1). To complete the proof, it will suffice to show that the equivalence $\operatorname{\mathcal{S}}_{< \mathbb {K} } \otimes _{ \mathbb {K} } \operatorname{\mathcal{S}}_{< \mathbb {K} } \simeq \operatorname{\mathcal{S}}_{< \mathbb {K} }$ of Example 9.7.4.9 exhibits $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ as a unit object of $\operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }}$, in the sense of Definition 2.1.2.5: that is, that the functors

\[ \operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }} \rightarrow \operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }} \quad \quad \operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{S}}_{< \mathbb {K} } \otimes _{\mathbb {K} } \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{S}}_{< \mathbb {K} } \]

are fully faithful. In fact, both of these functors are naturally isomorphic to the identity: see Proposition 9.7.4.8 and Remark 9.7.4.10. $\square$

Remark 9.7.7.15. The monoidal structure of Proposition 9.7.7.14 can be upgraded to a symmetric monoidal structure (Definition ), with the symmetry constraints supplied by Remark 9.7.3.13. We will later show that it can be upgraded further to a symmetric monoidal structure on the $\infty $-category $\operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K} -\mathrm{ccomp} }$ itself, before passing to the homotopy category: see Construction .

Exercise 9.7.7.16. In the situation of Proposition 9.7.7.14, suppose that we are given a smaller collection of simplicial sets $\mathbb {K}_0 \subseteq \mathbb {K}$. Show that the relative cocompletion functor

\[ \operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}_0-\mathrm{ccomp} }} \rightarrow \operatorname {h}\! \mathit{ \operatorname{\mathcal{QC}}_{< \lambda }^{ \mathbb {K}-\mathrm{ccomp} }} \quad \quad \operatorname{\mathcal{C}}\mapsto \widehat{\operatorname{\mathcal{C}}} \]

of Corollary 9.7.1.22 can be upgraded to a monoidal functor (see Definition 2.1.6.1), with tensor constraints given by the equivalences

\[ \widehat{\operatorname{\mathcal{C}}} \otimes _{ \mathbb {K} } \widehat{\operatorname{\mathcal{D}}} \simeq \widehat{ \operatorname{\mathcal{C}}\otimes _{ \mathbb {K}_0} \operatorname{\mathcal{D}}} \]

supplied by Proposition 9.7.5.3.