Remark 9.7.0.1. The preceding discussion makes sense in greater generality. Let $\mathbb {K}$ be a collection of simplicial sets, and let $\operatorname{\mathcal{A}}$ and $\operatorname{\mathcal{B}}$ be $\mathbb {K}$-cocomplete $\infty $-categories. We can then construct a $\mathbb {K}$-cocomplete tensor product $\operatorname{\mathcal{A}}\otimes _{ \mathbb {K} } \operatorname{\mathcal{B}}$, which is universal among $\mathbb {K}$-cocomplete $\infty $-categories $\operatorname{\mathcal{M}}$ which admit a functor $T: \operatorname{\mathcal{A}}\times \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{M}}$ which preserve $K$-indexed colimits separately in each variable, for each $K \in \mathbb {K}$.
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9.7 Tensor Products of $\infty $-Categories
Let $A$ and $B$ be abelian groups. If $M$ is another abelian group, we say that a function $T: A \times B \rightarrow M$ is bilinear if it satisfies the identities
\[ T(a, b+b' ) = T(a,b) + T(a,b') \quad \quad T(a+a', b) = T(a,b) + T(a', b). \]
Recall that, for fixed $A$ and $B$, the tensor product $A \otimes B$ is universal among abelian groups receiving a bilinear function with domain $A \times B$. More precisely, it is characterized (up to isomorphism) by the existence of a bilinear function
\[ T: A \times B \rightarrow A \otimes B \quad \quad (a,b) \mapsto a \otimes b \]
which satisfies the following universal property: for every abelian group $M$, composition with $T$ induces a bijection
\[ \{ \textnormal{Homomorphisms $A \otimes B \rightarrow M$} \} \rightarrow \{ \textnormal{Bilinear functions $A \times B \rightarrow M$} \} . \]
Our goal in this section is to develop a counterpart of the tensor product construction in the setting of higher category theory, where we replace abelian groups by (certain) $\infty $-categories, and the operation of addition by the formation of (certain) colimits. Let $\operatorname{\mathcal{A}}$ and $\operatorname{\mathcal{B}}$ be cocomplete $\infty $-categories. If $\operatorname{\mathcal{M}}$ is another cocomplete $\infty $-category, we say that a functor $T: \operatorname{\mathcal{A}}\times \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{M}}$ is separately cocontinuous if it satisfies the following pair of conditions:
For every object $A \in \operatorname{\mathcal{A}}$, the functor $T( A, - ): \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{M}}$ is cocontinuous.
For every object $B \in \operatorname{\mathcal{B}}$, the functor $T(-, B): \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{M}}$ is cocontinuous.
In §9.7.3, we show that there is a universal example of a separately cocontinuous functor
\[ T: \operatorname{\mathcal{A}}\times \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{A}}\widehat{\otimes } \operatorname{\mathcal{B}}\quad \quad (A,B) \mapsto A \otimes B, \]
which is characterized (up to equivalence) by the requirement that for every cocomplete $\infty $-category $\operatorname{\mathcal{M}}$, precomposition with $T$ induces an equivalence
\[ \{ \textnormal{Cocontinuous functors $\operatorname{\mathcal{A}}\widehat{\otimes } \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{M}}$} \} \rightarrow \{ \textnormal{Separately cocontinuous functors $\operatorname{\mathcal{A}}\times \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{M}}$} \} . \]
(Proposition 9.7.3.9). We will refer to $\operatorname{\mathcal{A}}\widehat{\otimes } \operatorname{\mathcal{B}}$ as the cocomplete tensor product of $\operatorname{\mathcal{A}}$ with $\operatorname{\mathcal{B}}$ (Notation 9.7.3.10). The existence of the cocomplete tensor product is a special case of a general result about relative cocompletions (Theorem 9.7.1.4), which we formulate and prove in §9.7.1 .
The cocomplete tensor product is manifestly symmetric in its arguments: for every pair of cocomplete $\infty $-categories $\operatorname{\mathcal{A}}$ and $\operatorname{\mathcal{B}}$, there is a canonical equivalence $\operatorname{\mathcal{A}}\widehat{\otimes } \operatorname{\mathcal{B}}\simeq \operatorname{\mathcal{B}}\widehat{\otimes } \operatorname{\mathcal{A}}$ (Remark 9.7.3.13). In §9.7.7, we show that it is also associative: if $\operatorname{\mathcal{C}}$ is another cocomplete $\infty $-category, there is a canonical equivalence
\[ \alpha : \operatorname{\mathcal{A}}\widehat{\otimes } ( \operatorname{\mathcal{B}}\widehat{\otimes } \operatorname{\mathcal{C}}) \quad \quad ( \operatorname{\mathcal{A}}\widehat{\otimes } \operatorname{\mathcal{B}}) \widehat{\otimes } \operatorname{\mathcal{C}} \]
which is characterized (up to isomorphism) by the existence of natural isomorphisms $\alpha (A \otimes (B \otimes C)) \xrightarrow {\sim } (A \otimes B) \otimes C$ (Proposition 9.7.7.1). Modulo size considerations, these equivalences furnish a monoidal structure on the homotopy category of cocomplete $\infty $-categories (Proposition 9.7.7.14). Here the role of the unit object is played by the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces, as we explain in §9.7.4 (Proposition 9.7.4.1).
The cocomplete tensor product is of particular interest in the presentable setting. In §9.7.5, we show that if $\operatorname{\mathcal{A}}$ and $\operatorname{\mathcal{B}}$ are presentable $\infty $-categories, then the cocomplete tensor product $\operatorname{\mathcal{A}}\widehat{\otimes } \operatorname{\mathcal{B}}$ is also presentable. In this case, the cocomplete tensor product admits a more explicit description: it can be identified with the $\infty $-category $\operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{A}}^{\operatorname{op}}, \operatorname{\mathcal{B}})$ of continuous functors from $\operatorname{\mathcal{A}}^{\operatorname{op}}$ to $\operatorname{\mathcal{B}}$ (Proposition 9.7.5.9). As a consequence, we deduce that the formation of cocomplete tensor products preserves Bousfield localization in each variable (Proposition 9.7.5.13). Some important examples of Bousfield localizations can be obtained in this way. In §9.7.6, we show that if $\operatorname{\mathcal{A}}$ is a presentable $\infty $-category, then the full subcategory $\operatorname{\mathcal{A}}^{\leq n} \subseteq \operatorname{\mathcal{A}}$ of $n$-truncated objects can be identified with the cocomplete tensor product $\operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{A}}$, where $\operatorname{\mathcal{S}}^{\leq n}$ is the $\infty $-category of $n$-truncated spaces (Proposition 9.7.6.5). It follows that the presentable $\infty $-category $\operatorname{\mathcal{S}}^{\leq n}$ is an idempotent with respect to the cocomplete tensor product (Definition 9.7.8.1), a phenomenon which we study more generally in §9.7.8.
Structure
- Subsection 9.7.1: Relative Cocompletion
- Subsection 9.7.2: Digression: Sifted $\infty $-Categories
- Subsection 9.7.3: Tensor Products of $\infty $-Categories
- Subsection 9.7.4: Unitality
- Subsection 9.7.5: Tensor Products of Presentable $\infty $-Categories
- Subsection 9.7.6: Locally Truncated $\infty $-Categories
- Subsection 9.7.7: Associativity
- Subsection 9.7.8: Idempotent $\infty $-Categories