Proposition 9.7.6.1. Let $\kappa $ be a regular cardinal, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are $\kappa $-cocomplete, and let $n$ be an integer. If either $\operatorname{\mathcal{C}}$ or $\operatorname{\mathcal{D}}$ is locally $n$-truncated, then the tensor product $\operatorname{\mathcal{D}}\otimes _{\kappa } \operatorname{\mathcal{C}}$ is locally $n$-truncated.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.7.6 Locally Truncated $\infty $-Categories
Let $n$ be an integer. Recall that an $\infty $-category $\operatorname{\mathcal{C}}$ is locally $n$-truncated if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $n$-truncated (Definition 4.7.6.1). Our starting point is the following elementary observation:
Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{D}}$ is locally $n$-truncated. We wish to show that every object of $X \in \operatorname{\mathcal{D}}\otimes _{\kappa } \operatorname{\mathcal{C}}$ is $n$-cotruncated (Remark 4.7.6.2). Since the collection of cotruncated objects of $\operatorname{\mathcal{D}}\otimes _{\kappa } \operatorname{\mathcal{C}}$ is closed under colimits (Corollary 7.3.8.7), we may assume without loss of generality that $X = D \otimes C$, for some objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$ (Remark 9.7.3.11). Our assumption on $\operatorname{\mathcal{D}}$ guarantees that the object $D \in \operatorname{\mathcal{D}}$ is $n$-cotruncated. The desired result now follows from the observation that the functor
\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}\otimes _{\kappa } \operatorname{\mathcal{C}}\quad \quad D \mapsto D \otimes C \]
is $\kappa $-cocontinuous, and therefore carries $n$-cotruncated objects to $n$-cotruncated objects (Remark 7.1.4.10). $\square$
Recall that, if $\kappa $ is an uncountable regular cardinal, then $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa }$ denotes the $\infty $-categories of spaces which are $n$-truncated and essentially $\kappa $-small (Variant 6.2.2.13). We have the following variant of Proposition 9.7.4.8:
Proposition 9.7.6.2. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be a $\kappa $-cocomplete $\infty $-category, and let $n$ be an integer. Then:
The tensor product $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa } \otimes _{\kappa } \operatorname{\mathcal{C}}$ is locally $n$-truncated.
If $\operatorname{\mathcal{D}}$ is a $\kappa $-cocomplete $\infty $-category which is locally $n$-truncated, then precomposition with the map
induces an equivalence of $\infty $-categories
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{\leq n}_{< \kappa } \otimes _{\kappa } \operatorname{\mathcal{C}}\quad \quad C \mapsto \Delta ^0 \otimes C \]
\[ \theta : \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{S}}^{\leq n}_{< \kappa } \otimes _{\kappa } \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}). \]
Stated more informally, Proposition 9.7.6.2 asserts that the tensor product $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa } \otimes _{\kappa } \operatorname{\mathcal{C}}$ is universal among $\kappa $-cocomplete locally $n$-truncated $\infty $-categories which admit a $\kappa $-cocontinuous functor from $\operatorname{\mathcal{C}}$.
Proof of Proposition 9.7.6.2. Assertion $(1)$ is a special case of Proposition 9.7.6.1. To prove $(2)$, let $\operatorname{\mathcal{D}}$ be a $\kappa $-cocomplete $\infty $-category which is locally $n$-truncated. Invoking the universal property of the tensor product (and using Remark 9.7.3.3), we can identify $\theta $ with the functor
\[ \operatorname{Fun}^{ \kappa -\mathrm{cocont} }( \operatorname{\mathcal{S}}^{\leq n}_{< \kappa }, \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]
given by evaluation at $\Delta ^0$. Since the $\infty $-category $\operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is locally $n$-truncated (Corollary 4.8.4.14), this is a special case of Corollary 8.4.4.3. $\square$
Corollary 9.7.6.3. Let $\kappa $ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be a $\kappa $-cocomplete $\infty $-category. Then $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if the functor is an equivalence of $\infty $-categories.
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{\leq n}_{< \kappa } \otimes _{\kappa } \operatorname{\mathcal{C}}\quad \quad C \mapsto \Delta ^0 \otimes C \]
Example 9.7.6.4. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be a cocomplete $\infty $-category. Then:
The cocomplete tensor product $\operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{C}}$ is locally $n$-truncated.
The $\infty $-category $\operatorname{\mathcal{D}}$ is locally $n$-truncated if and only if the functor
is an equivalence of $\infty $-categories.
For every cocomplete $\infty $-category $\operatorname{\mathcal{D}}$ which is locally $n$-truncated, precomposition with $F$ induces an equivalence of $\infty $-categories
\[ F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{C}}\quad \quad C \mapsto \Delta ^0 \otimes C \]
\[ \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}). \]
These assertions follow by applying Proposition 9.7.6.2 (and Corollary 9.7.6.3) in the special case where $\kappa = \Omega $ is a strongly inaccessible cardinal.
In the setting of presentable $\infty $-categories, the cocomplete tensor product $\operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{C}}$ admits a more concrete description.
Proposition 9.7.6.5. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. Then the functor admits a fully faithful right adjoint $\operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$, whose essential image is the full subcategory of $\operatorname{\mathcal{C}}^{\leq n} \subseteq \operatorname{\mathcal{C}}$ spanned by the $n$-truncated objects.
\[ F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{C}}\quad \quad X \mapsto \Delta ^{0} \otimes X \]
Stated more informally, Proposition 9.7.6.5 supplies a canonical equivalence of $\infty $-categories $\operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}^{\leq n}$.
Proof of Proposition 9.7.6.5. Since $\operatorname{\mathcal{C}}$ is presentable, the covariant Yoneda embedding determines an equivalence of $\infty $-categories $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. Let $h_{\bullet }^{-1}$ denote a homotopy inverse to $h_{\bullet }$. Using Proposition 9.7.5.9, we can identify the tensor product $\operatorname{\mathcal{S}}^{\leq n} \widehat{\otimes } \operatorname{\mathcal{C}}$ with the $\infty $-category $\operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{\leq n} )$. Under this identification, the right adjoint to $F$ corresponds to the composition
\[ \operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{\leq n} ) \hookrightarrow \operatorname{Fun}^{\operatorname{\mathrm{cont}}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow {h_{\bullet }^{-1}} \operatorname{\mathcal{C}} \]
(see Example 9.7.5.11 and Remark 9.7.5.12). It follows that the right adjoint of $F$ is fully faithful, and its essential image consists of those objects $X \in \operatorname{\mathcal{C}}$ for which the representable functor $h_{X}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ factors through $\operatorname{\mathcal{S}}^{\leq n}$, which is a restatement of the condition that $X$ is $n$-truncated. $\square$
Remark 9.7.6.6 (The Universal Property of Truncation). Let $n$ be an integer, let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category, and let $\operatorname{\mathcal{C}}^{\leq n}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $n$-truncated objects. It follows from Example 9.5.6.18 that $\operatorname{\mathcal{C}}^{\leq n}$ is a Bousfield localization of $\operatorname{\mathcal{C}}$; in particular, the inclusion functor $\operatorname{\mathcal{C}}^{\leq n} \subseteq \operatorname{\mathcal{C}}$ admits a left adjoint $\tau _{\leq n}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\leq n}$. Combining Proposition 9.7.6.5 with Example 9.7.6.4, we see that $\tau _{\leq n}$ is universal among cocontinuous functors from $\operatorname{\mathcal{C}}$ to presentable $\infty $-categories which are locally $n$-truncated. More precisely, if $\operatorname{\mathcal{D}}$ is a presentable $\infty $-category which is locally $n$-truncated, then precomposition with $\tau _{\leq n}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}^{\leq n}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
Example 9.7.6.7. Let $\operatorname{Set}$ denote the category of sets. Then $\operatorname{N}_{\bullet }( \operatorname{Set})$ is a presentable $\infty $-category, which is equivalent to the full subcategory $\operatorname{\mathcal{S}}^{\leq 0} \subseteq \operatorname{\mathcal{S}}$ spanned by the $0$-truncated Kan complexes (see Remark 3.1.6.6 and Example 4.7.4.17). For every presentable $\infty $-category $\operatorname{\mathcal{C}}$, Proposition 9.7.6.5 (applied in the special case $n = 0$) supplies an equivalence of $\infty $-categories $\operatorname{N}_{\bullet }( \operatorname{Set}) \widehat{\otimes } \operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}^{\heartsuit }$, where $\operatorname{\mathcal{C}}^{\heartsuit } \subseteq \operatorname{\mathcal{C}}$ is the full subcategory spanned by the discrete objects.
Example 9.7.6.8. The standard $1$-simplex $\Delta ^1$ is a presentable $\infty $-category, which is equivalent to the full subcategory $\operatorname{\mathcal{S}}^{\leq -1} \subseteq \operatorname{\mathcal{S}}$ spanned by the $(-1)$-truncated Kan complexes (see Example 4.7.4.17). For every presentable $\infty $-category $\operatorname{\mathcal{C}}$, Proposition 9.7.6.5 (applied in the special case $n = -1$) supplies an equivalence $\Delta ^1 \widehat{\otimes } \operatorname{\mathcal{C}}\simeq \operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$, where $\operatorname{Sub}(\operatorname{\mathcal{C}})$ denotes the (partially ordered) set of isomorphism classes of subterminal objects of $\operatorname{\mathcal{C}}$ (see Notation 4.7.4.9).
Example 9.7.6.9. The standard $0$-simplex $\Delta ^0$ is a presentable $\infty $-category, which is equivalent to the full subcategory $\operatorname{\mathcal{S}}^{\leq -2} \subseteq \operatorname{\mathcal{S}}$ spanned by the contractible Kan complexes. For every presentable $\infty $-category $\operatorname{\mathcal{C}}$, Proposition 9.7.6.5 (applied in the special case $n = -2$) supplies an equivalence of $\Delta ^0 \widehat{\otimes } \operatorname{\mathcal{C}}$ with the full subcategory $\operatorname{\mathcal{C}}^{\leq -2} \subseteq \operatorname{\mathcal{C}}$ spanned by the final objects, which is also a contractible Kan complex (Corollary 4.7.3.14).
Proposition 9.7.6.2 has a counterpart in the case $\kappa = \aleph _0$. For every integer $n$, we let $\operatorname{\mathcal{S}}^{\leq n}_{\mathrm{fin}}$ denote the full subcategory of $\operatorname{\mathcal{S}}$ spanned by the $n$-truncations of essentially finite Kan complexes (see Notation 9.4.3.6).
Variant 9.7.6.10. Let $\operatorname{\mathcal{C}}$ be a finitely cocomplete $\infty $-category and let $n$ be an integer. Then:
The tensor product $\operatorname{\mathcal{S}}^{\leq n}_{\mathrm{fin}} \otimes _{\aleph _0} \operatorname{\mathcal{C}}$ is locally $n$-truncated.
If $\operatorname{\mathcal{D}}$ is a finitely cocomplete $\infty $-category which is locally $n$-truncated, then precomposition with the map
induces an equivalence of $\infty $-categories $\theta : \operatorname{Fun}^{\operatorname{rex}}( \operatorname{\mathcal{S}}^{\leq n}_{\mathrm{fin}} \otimes _{\aleph _0} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\operatorname{rex}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
\[ F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{\leq n}_{ \mathrm{fin} } \otimes _{\kappa } \operatorname{\mathcal{C}}\quad \quad C \mapsto \Delta ^0 \otimes C \]
In particular, $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if $F$ is an equivalence of $\infty $-categories.
Proof. Arguing as in the proof of Proposition 9.7.6.2, we are reduced to showing that evaluation on $\Delta ^0$ induces an equivalence of $\infty $-categories
\[ \operatorname{Fun}^{\operatorname{rex}}( \operatorname{\mathcal{S}}^{\leq n}_{\mathrm{fin}}, \operatorname{Fun}^{\operatorname{rex}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}^{\operatorname{rex}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}), \]
which is a special case of Corollary 9.5.5.12. $\square$