Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 3.1.6.1). In this section, we show that $\operatorname{\mathcal{S}}$ plays the role of a unit object with respect to the cocomplete tensor product $\widehat{\otimes }$ of Variant 9.7.3.15. More precisely, we have the following:
Proposition 9.7.4.1. Let $\operatorname{\mathcal{C}}$ be a cocomplete $\infty $-category. Then the functor is an equivalence of $\infty $-categories.
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\widehat{\otimes } \operatorname{\mathcal{C}}\quad \quad X \mapsto \Delta ^0 \otimes X \]
Proposition 9.7.4.1 is a special case of a more general assertion.
Notation 9.7.4.2. Let $\mathbb {K}$ be a collection of simplicial sets. Choose an uncountable regular cardinal $\kappa $ such that each $K \in \mathbb {K}$ is $\kappa $-small, so that the $\infty $-category $\operatorname{\mathcal{S}}_{ < \kappa }$ is $\mathbb {K}$-cocomplete (Corollary 7.4.3.8). We let $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ denote the smallest full subcategory of $\operatorname{\mathcal{S}}_{< \kappa }$ which contains the standard $0$-simplex $\Delta ^0$ and is closed under $K$-indexed colimits, for each $K \in \mathbb {K}$.
Example 9.7.4.3. In the situation of Notation 9.7.4.2, suppose that $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets. Then $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ coincides with $\operatorname{\mathcal{S}}_{< \kappa }$. See Example 7.1.2.10.
Example 9.7.4.4. If $\mathbb {K}$ is the collection of all small simplicial sets, then $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ can be identified with the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ (Construction 3.1.6.1). This follows by applying Example 9.7.4.3 in the special case where $\kappa = \Omega $ is a strongly inaccessible cardinal.
Example 9.7.4.5. Let $\mathbb {K}$ be the collection of all finite simplicial sets. Then $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ can be identified with the full subcategory $\operatorname{\mathcal{S}}_{ \mathrm{fin} } \subset \operatorname{\mathcal{S}}$ spanned by the essentially finite Kan complexes (see Definition 9.2.6.1). This is a reformulation of Proposition 9.2.6.3.
Example 9.7.4.6. Let $\mathbb {K}$ be a collection of weakly contractible simplicial sets. Then $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ can be identified with the full subcategory of $\operatorname{\mathcal{S}}$ spanned by the contractible Kan complexes.
Remark 9.7.4.7. Let $\mathbb {K}$ be a collection of simplicial sets. Then the inclusion $\{ \Delta ^0 \} \hookrightarrow \operatorname{\mathcal{S}}_{< \mathbb {K} }$ exhibits $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ as a $\mathbb {K}$-cocompletion of the contractible Kan complex $\{ \Delta ^0 \} $. In other words, for every $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{C}}$, the evaluation functor is an equivalence of $\infty $-categories. This is a special case of Proposition 8.4.6.8.
\[ \operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \operatorname{\mathcal{S}}_{< \mathbb {K} }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad F \mapsto F( \Delta ^0 ) \]
Proposition 9.7.4.1 is a consequence of the following:
Proposition 9.7.4.8. Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{C}}$ be a $\mathbb {K}$-cocomplete $\infty $-category. Then:
There is an (essentially unique) $\mathbb {K}$-bilinear functor $T: \operatorname{\mathcal{S}}_{< \mathbb {K} } \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ for which the restriction $T( \Delta ^0, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$.
The functor $T$ exhibits $\operatorname{\mathcal{C}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ with $\operatorname{\mathcal{C}}$.
Proof. Using Remark 9.7.3.3, we can reformulate $(1)$ as follows:
There is an (essentially unique) $\mathbb {K}$-cocontinuous functor $Q: \operatorname{\mathcal{S}}_{< \mathbb {K} } \rightarrow \operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ such that $Q( \Delta ^0 )$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$.
This follows from the universal property of Remark 9.7.4.7. To prove $(2)$, let $\operatorname{\mathcal{D}}$ be another $\mathbb {K}$-cocomplete $\infty $-category. Using Remarks 9.7.3.3 and 9.7.4.7 again, we are reduced to proving that the functor
is an equivalence of $\infty $-categories. Unwinding the definitions, we see that this functor is given by precomposition with $Q( \Delta ^0 )$, and is therefore isomorphic to the identity. $\square$
Example 9.7.4.9. Let $\mathbb {K}$ be a collection of simplicial sets. Fix an uncountable regular cardinal $\kappa $ such that each $K \in \mathbb {K}$ is $\kappa $-small. Applying Example 9.7.3.4, we see that the formation of cartesian products defines a $\kappa $-bilinear functor $\operatorname{\mathcal{S}}_{< \kappa } \times \operatorname{\mathcal{S}}_{< \kappa } \rightarrow \operatorname{\mathcal{S}}_{ < \kappa }$. It follows that, if $X$ and $Y$ belong the the subcategory $\operatorname{\mathcal{S}}_{< \mathbb {K} } \subseteq \operatorname{\mathcal{S}}_{< \kappa }$, then the product $X \times Y$ also belongs to $\operatorname{\mathcal{S}}_{< \mathbb {K} }$. We therefore obtain a $\mathbb {K}$-bilinear functor which satisfies the hypotheses of Proposition 9.7.4.8, and therefore induces an equivalence of $\infty $-categories $\operatorname{\mathcal{S}}_{< \mathbb {K}} \otimes _{ \mathbb {K} } \operatorname{\mathcal{S}}_{ < \mathbb {K} } \xrightarrow {\sim } \operatorname{\mathcal{S}}_{< \mathbb {K} }$.
\[ \operatorname{\mathcal{S}}_{< \mathbb {K}} \times \operatorname{\mathcal{S}}_{< \mathbb {K} } \rightarrow \operatorname{\mathcal{S}}_{ < \mathbb {K} } \quad \quad (X,Y) \mapsto X \times Y \]
Remark 9.7.4.10 (Functoriality). Let $\mathbb {K}$ be a collection of simplicial sets and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a $\mathbb {K}$-cocontinuous functor between $\mathbb {K}$-cocomplete $\infty $-categories. Then the diagram commutes up to (canonical) isomorphism, where $T_{\operatorname{\mathcal{C}}}$ and $T_{\operatorname{\mathcal{D}}}$ are the $\mathbb {K}$-bilinear functors obtained by applying Proposition 9.7.4.8 to $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively.
\[ \xymatrix { \operatorname{\mathcal{S}}_{< \mathbb {K} } \times \operatorname{\mathcal{C}}\ar [r]^{T_{\operatorname{\mathcal{C}}}} \ar [d]^{\operatorname{id}\times F} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{S}}_{< \mathbb {K} } \times \operatorname{\mathcal{D}}\ar [r]^{T_{\operatorname{\mathcal{D}}}} & \operatorname{\mathcal{D}}} \]
Remark 9.7.4.11 (Functoriality of Copowers). In the situation of Proposition 9.7.4.8, the functor carries each pair $(S, X) \in \operatorname{\mathcal{S}}_{< \mathbb {K} } \times \operatorname{\mathcal{C}}$ to a copower of $X$ by $S$, in the sense of Notation 7.1.2.5. To see this, fix an uncountable regular cardinal $\kappa $ such that each $K \in \mathbb {K}$ is $\kappa $-small. Using Corollary 8.3.3.17, we can enlarge $\operatorname{\mathcal{C}}$ to guarantee that it is $\kappa $-cocomplete, so that $T$ extends to a $\kappa $-bilinear functor $\widehat{T}: \operatorname{\mathcal{S}}_{< \kappa } \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (Proposition 9.7.4.8). In this case, the result follows from Remark 8.4.4.2.
\[ T: \operatorname{\mathcal{S}}_{< \mathbb {K} } \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}} \]
Corollary 9.7.4.12. Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\mathbb {K}$-cocomplete. Then the functor is an equivalence of $\infty $-categories.
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mathbb {K} } \otimes _{ \mathbb {K} } \operatorname{\mathcal{C}}\quad \quad X \mapsto \Delta ^0 \otimes X \]
Proof. Choose a $\mathbb {K}$-bilinear functor $T: \operatorname{\mathcal{S}}_{< \mathbb {K} } \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocomplete tensor product of $\operatorname{\mathcal{S}}_{< \mathbb {K} }$ with $\operatorname{\mathcal{C}}$; we wish to show that the restriction $T|_{ \{ \Delta ^0 \} \times \operatorname{\mathcal{C}}}$ is an equivalence of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. By virtue of Proposition 9.7.4.8, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \operatorname{\mathcal{D}}$ and that $F$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$. $\square$
Proof of Proposition 9.7.4.1. Apply Corollary 9.7.4.12 in the special case where $\mathbb {K}$ is the collection of small simplicial sets. $\square$