Definition 5.4.5.1. Let $\kappa $ be an uncountable cardinal. We will say that a simplicial set $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small if there exists a categorical equivalence of simplicial sets $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is a $\kappa $-small $\infty $-category.
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5.4.5 Essential Smallness
Let $\kappa $ be an infinite cardinal. Beware that the condition that a simplicial set is $\kappa $-small is not invariant under categorical equivalence. For this reason, it is useful to consider the following variant of Definition 5.4.4.1:
Remark 5.4.5.2. Let $\kappa $ be an uncountable cardinal, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a categorical equivalence of simplicial sets. Then $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small if and only if $\operatorname{\mathcal{D}}$ is essentially $\kappa $-small.
Remark 5.4.5.3. Let $\kappa $ be an uncountable cardinal. Then a simplicial set $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small if and only if the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is essentially $\kappa $-small. See Remark 5.4.4.3.
Variant 5.4.5.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set. We say that $\operatorname{\mathcal{C}}$ is essentially small if there exists a categorical equivalence $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is a small $\infty $-category.
Proposition 5.4.5.5. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be a $\kappa $-small simplicial set. Then there exists an inner anodyne morphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is a $\kappa $-small $\infty $-category. In particular, $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small.
Proof. Without loss of generality, we may assume that $\kappa $ is the least uncountable cardinal for which $\operatorname{\mathcal{C}}$ is $\kappa $-small, so that $\kappa $ is regular (Remark 5.4.4.7). We proceed as in the proof of Proposition 4.1.3.2. We will construct $\operatorname{\mathcal{D}}$ as the colimit of a diagram of inner anodyne morphisms
\[ \operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}(0) \hookrightarrow \operatorname{\mathcal{C}}(1) \hookrightarrow \operatorname{\mathcal{C}}(2) \hookrightarrow \operatorname{\mathcal{C}}(3) \hookrightarrow \cdots \]
where each transition map fits into a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \coprod _{s \in S(n)} \Lambda ^{n_ s}_{i_ s} \ar [d] \ar [r]^-{ \{ u_ s \} _{s \in S(n)} } & \operatorname{\mathcal{C}}(n) \ar [d] \\ \coprod _{s \in S(n)} \Delta ^{n_ s} \ar [r] & \operatorname{\mathcal{C}}(n+1); } \]
here the coproducts are indexed by the collection $\{ u_{s}: \Lambda ^{n_ s}_{i_ s} \rightarrow \operatorname{\mathcal{C}}(n) \} _{s \in S(n)}$ of all inner horns in the simplicial set $\operatorname{\mathcal{C}}(n)$. Note that if the simplicial set $\operatorname{\mathcal{C}}(n)$ is $\kappa $-small, then the set $S(n)$ is also $\kappa $-small (Proposition 5.4.4.9), so that $\operatorname{\mathcal{C}}(n+1)$ is also $\kappa $-small. Since $\kappa $ is regular and uncountable, it follows that the colimit $\operatorname{\mathcal{C}}= \varinjlim \operatorname{\mathcal{C}}(n)$ is $\kappa $-small (Remark 5.4.4.6). $\square$
Warning 5.4.5.6. The statement of Proposition 5.4.5.5 is false in the case $\kappa = \aleph _0$. If $S$ is a finite simplicial set, we generally cannot choose a categorical equivalence $f: S \rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is an $\infty $-category which is also a finite simplicial set. For example, take $S = \Delta ^2 / \operatorname{\partial \Delta }^{2}$, so that the geometric realization $| S |$ is homeomorphic to a sphere of dimension $2$. Since every edge of $S$ is degenerate, the homotopy category $\mathrm{h} \mathit{S}$ is a groupoid. Consequently, if $f$ is a categorical equivalence from $S$ to an $\infty $-category $\operatorname{\mathcal{D}}$, then $\operatorname{\mathcal{D}}$ is a Kan complex (Proposition 4.4.2.1), which is homotopy equivalent to the singular simplicial set $\operatorname{Sing}_{\bullet }( | S |)$ (Theorem 3.5.4.1). It follows that $\pi _{2}(\operatorname{\mathcal{D}})$ is an infinite cyclic group (generated by the homotopy class $[f]$), so that the Kan complex $\operatorname{\mathcal{D}}$ must contain infinitely many $2$-simplices.
Remark 5.4.5.7 (Coproducts). Let $\kappa $ be an uncountable cardinal and let $\{ \operatorname{\mathcal{C}}_{i} \} _{i \in I}$ be a collection of essentially $\kappa $-small simplicial sets. Suppose that the cardinality of the index set $I$ is smaller than the cofinality $\mathrm{cf}(\kappa )$. Then the coproduct $\coprod _{i \in I} \operatorname{\mathcal{C}}_ i$ is also essentially $\kappa $-small. This follows by combining Remark 5.4.4.5 with Corollary 4.5.3.10. In particular:
The collection of essentially $\kappa $-small simplicial sets is closed under finite coproducts.
If $\kappa $ is regular, then the collection of essentially $\kappa $-small simplicial sets is closed under $\kappa $-small coproducts.
Remark 5.4.5.8 (Products). Let $\kappa $ be an uncountable cardinal and let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a finite collection of simplicial sets which are essentially $\kappa $-small. Then the product $\prod _{i \in I} \operatorname{\mathcal{C}}_ i$ is essentially $\kappa $-small. This follows by combining Corollary 5.4.4.14, since the collection of categorical equivalences is stable under the formation of finite products (Remark 4.5.3.7).
Variant 5.4.5.9. Let $\kappa $ be an uncountable cardinal and let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a collection of essentially $\kappa $-small $\infty $-categories. Suppose that the cardinality of the index set $I$ has smaller than the exponential cofinality $\mathrm{ecf}(\kappa )$. Then the product $\prod _{i \in I} \operatorname{\mathcal{C}}_{i}$ is also essentially $\kappa $-small. This follows by combining Corollary 5.4.4.14 with Remark 4.5.1.17.
Remark 5.4.5.10. Let $\lambda $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small, and let $K$ be a simplicial set. Suppose that $K$ is $\kappa $-small, where $\kappa = \mathrm{ecf}(\lambda )$ is the exponential cofinality of $\lambda $. Then the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is essentially $\lambda $-small. To prove this, we can use Remark 4.5.1.16 to reduce to the case where $\operatorname{\mathcal{C}}$ is $\lambda $-small, in which case it follows from Corollary 5.4.4.12. Moreover, if $\kappa $ is uncountable, then it suffices to assume that $K$ is essentially $\kappa $-small.
Proposition 5.4.5.11. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small. Then any replete subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is also essentially $\kappa $-small.
Proof. Choose an equivalence of $\infty $-categories $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{D}}$ is $\kappa $-small. Then the inverse image $\operatorname{\mathcal{D}}_0 = F^{-1}( \operatorname{\mathcal{C}}_0 )$ is $\kappa $-small (Remark 5.4.4.8), and the functor $F$ restricts to an equivalence of $\infty $-categories $\operatorname{\mathcal{D}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ (Corollary 4.5.2.23). $\square$
Corollary 5.4.5.12. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small. Then the core $\operatorname{\mathcal{C}}^{\simeq }$ is an essentially $\kappa $-small Kan complex.
Proof. Since $\operatorname{\mathcal{C}}^{\simeq }$ is a replete subcategory of $\operatorname{\mathcal{C}}$ (Proposition 4.4.3.6), this is a special case of Proposition 5.4.5.11. $\square$
Corollary 5.4.5.13. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small. Then any full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is essentially $\kappa $-small.
Proof. Let $\operatorname{\mathcal{C}}_1 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects $X \in \operatorname{\mathcal{C}}$ which are isomorphic to an object of $\operatorname{\mathcal{C}}_0$. Proposition 5.4.5.11 guarantees that $\operatorname{\mathcal{C}}_1$ is essentially $\kappa $-small. Since the inclusion $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}_1$ is an equivalence of $\infty $-categories, it follows that $\operatorname{\mathcal{C}}_0$ is also essentially $\kappa $-small (Remark 5.4.5.2). $\square$
Proposition 5.4.5.14. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $\kappa $ be an uncountable cardinal. If $\operatorname{\mathcal{C}}_0$, $\operatorname{\mathcal{C}}_1$, and $\operatorname{\mathcal{C}}$ are essentially $\kappa $-small, then the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is also essentially $\kappa $-small.
Proof. Choose equivalences of $\infty $-categories
\[ \operatorname{\mathcal{D}}_0 \rightarrow \operatorname{\mathcal{C}}_0 \quad \quad \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}\quad \quad \operatorname{\mathcal{D}}_1 \rightarrow \operatorname{\mathcal{C}}_1, \]
where $\operatorname{\mathcal{D}}_0$, $\operatorname{\mathcal{D}}_1$, and $\operatorname{\mathcal{D}}$ are $\kappa $-small. By virtue of Remark 4.6.4.4, the induced maps
\[ \operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \leftarrow \operatorname{\mathcal{D}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_1 \rightarrow \operatorname{\mathcal{D}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1 \]
are equivalences of $\infty $-categories. It will therefore suffice to show that the $\infty $-category $\operatorname{\mathcal{D}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1$ is $\kappa $-small. This follows from Corollaries 5.4.4.12 and 5.4.4.11, since $\operatorname{\mathcal{D}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1$ can be identified with a simplicial subset of the product $\operatorname{\mathcal{D}}_0 \times \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}}) \times \operatorname{\mathcal{D}}_1$. $\square$
Corollary 5.4.5.15. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $\kappa $ be an uncountable cardinal. If $\operatorname{\mathcal{C}}_0$, $\operatorname{\mathcal{C}}_1$, and $\operatorname{\mathcal{C}}$ are essentially $\kappa $-small, then the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is essentially $\kappa $-small.
Proof. Since $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a full subcategory of the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$, this follows from Proposition 5.4.5.14 and Corollary 5.4.5.13. $\square$
Corollary 5.4.5.16. Let $\kappa $ be an uncountable cardinal and suppose we are given a categorical pullback diagram of $\infty $-categories If $\operatorname{\mathcal{C}}_0$, $\operatorname{\mathcal{C}}$, and $\operatorname{\mathcal{C}}_1$ are essentially $\kappa $-small, then $\operatorname{\mathcal{C}}_{01}$ is essentially $\kappa $-small.
5.34
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-essentially-small} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
Proof. Combine Remark 5.4.5.2 with Corollary 5.4.5.15. $\square$