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Proposition 5.4.8.8. Let $\kappa $ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small.

$(2)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $\kappa $-small and the set of isomorphism classes $\pi _0( \operatorname{\mathcal{C}}^{\simeq } )$ is $\kappa $-small.

$(3)$

The Kan complex $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$ is essentially $\kappa $-small.

$(4)$

For every finite simplicial set $K$, the Kan complex $\operatorname{Fun}(K, \operatorname{\mathcal{C}})^{\simeq }$ is essentially $\kappa $-small.

$(5)$

For every integer $n \geq 0$, the set $\pi _0( \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}})^{\simeq } )$ is $\kappa $-small. Moreover, for every map $b: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$, the fundamental group $\pi _1( \operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})^{\simeq }, b)$ is $\kappa $-small.

Proof. The implication $(1) \Rightarrow (2)$ follows from Example 5.4.8.4. We next show that $(2) \Rightarrow (3)$. Assume that condition $(2)$ is satisfied; we wish to show that the Kan complex $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$ is essentially $\kappa $-small. Corollary 4.4.5.4 implies that the restriction map

\[ \theta : \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^{1}, \operatorname{\mathcal{C}})^{\simeq } \simeq \operatorname{\mathcal{C}}^{\simeq } \times \operatorname{\mathcal{C}}^{\simeq } \]

is a Kan fibration. Moreover, for each vertex $(X,Y) \in \operatorname{\mathcal{C}}^{\simeq } \times \operatorname{\mathcal{C}}^{\simeq }$, the fiber $\theta ^{-1} \{ (X,Y) \} $ can be identified with the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, which is essentially $\kappa $-small by virtue of $(2)$. Using Corollary 5.4.7.2 (and Remark 5.4.5.8), we are reduced to proving that the Kan complex $\operatorname{\mathcal{C}}^{\simeq }$ is essentially $\kappa $-small. Fix a vertex $X \in \operatorname{\mathcal{C}}^{\simeq }$. For $n \geq 2$, Example 4.6.1.13 supplies an isomorphism $\pi _{n}( \operatorname{\mathcal{C}}^{\simeq }, X) \simeq \pi _{n-1}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X), \operatorname{id}_ X)$, so that the homotopy group $\pi _{n}( \operatorname{\mathcal{C}}^{\simeq }, X)$ is essentially small by virtue of assumption $(2)$. Similarly, the fundamental group $\pi _{1}( \operatorname{\mathcal{C}}^{\simeq }, X)$ can be identified with the subset of $\pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) )$ spanned by the homotopy classes of isomorphisms, which is also $\kappa $-small. Since $\pi _0( \operatorname{\mathcal{C}}^{\simeq } )$ is $\kappa $-small by virtue of assumption $(2)$, Proposition 5.4.7.1 implies that the Kan complex $\operatorname{\mathcal{C}}^{\simeq }$ is essentially $\kappa $-small.

We now show that $(3)$ implies $(4)$. Assume that the Kan complex $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$ is essentially $\kappa $-small and let $K$ be a finite simplicial set; we wish to show that $\operatorname{Fun}(K, \operatorname{\mathcal{C}})^{\simeq }$ is also essentially $\kappa $-small. We proceed by induction on the dimension $n$ of $K$ and the number of nondegenerate $n$-simplices of $K$. If $K$ is empty, there is nothing to prove. Otherwise, there exists a pushout square of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \Delta ^ n \ar [d] \\ K' \ar [r] & K. } \]

Since the horizontal maps are monomorphisms, this diagram is also a categorical pushout square (Example 4.5.4.12) and therefore induces a homotopy pullback diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}})^{\simeq } \ar [l] \\ \operatorname{Fun}( K', \operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}( K, \operatorname{\mathcal{C}})^{\simeq } \ar [l] \ar [u] . } \]

Our inductive hypothesis guarantees that $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{C}})^{\simeq }$ and $\operatorname{Fun}(K', \operatorname{\mathcal{C}})^{\simeq }$ are essentially $\kappa $-small. It will therefore suffice to show that the Kan complex $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}})^{\simeq }$ is essentially $\kappa $-small (Corollary 5.4.5.16). If $n=1$, this follows from assumption $(3)$. If $n \geq 2$, then the inclusion map $\Lambda ^{n}_{1} \hookrightarrow \Delta ^ n$ induces a homotopy equivalence $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( \Lambda ^{n}_{1},\operatorname{\mathcal{C}})^{\simeq }$, so that the desired result again follows from our inductive hypothesis. It will therefore suffice to treat the case $n=0$: that is, to show that the Kan complex $\operatorname{\mathcal{C}}^{\simeq }$ is essentially $\kappa $-small. This follows from Corollary 5.4.5.13, since $\operatorname{\mathcal{C}}^{\simeq }$ is homotopy equivalent to the summand $\operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq } \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$ (see Corollary 4.4.5.10).

The implication $(4) \Rightarrow (5)$ follows from Proposition 5.4.7.1. We will complete the proof by showing that $(5)$ implies $(1)$. Assume that condition $(5)$ is satisfied; we will show that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. We now proceed as in the proof of Proposition 5.4.7.1. Using Proposition 5.4.6.12, we can reduce to the case where $\operatorname{\mathcal{C}}$ is minimal. In this case, we wish to show that $\operatorname{\mathcal{C}}$ is $\kappa $-small. By virtue of Proposition 5.4.4.9, it will suffice to show that the collection of $n$-simplices of $\operatorname{\mathcal{C}}$ is $\kappa $-small, for each $n \geq 0$. Our proof proceeds by induction on $n$. Using our inductive hypothesis (together with Remark 5.4.3.4 and Proposition 5.4.3.5), we see that the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})$ is $\kappa $-small. Since $\kappa $ is regular, it will suffice to show that each fiber of the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})$ is $\kappa $-small.

Set $E = \operatorname{Fun}( \Delta ^{n}, \operatorname{\mathcal{C}})^{\simeq }$ and $B = \operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{C}})^{\simeq }$, so that the inclusion map $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^ n$ induces a Kan fibration $q: E \rightarrow B$ (Corollary 4.4.5.4). Fix a vertex $b \in B$ and set $E_{b} = \{ b\} \times _{B} E$; we wish to show that the the set of vertices of $E_{b}$ is $\kappa $-small. Since $\operatorname{\mathcal{C}}$ is minimal, each vertex of $E_{b}$ belongs to a different connected component. It will therefore suffice to show that the set of connected components $\pi _0(E_ b)$ is $\kappa $-small. Assumption $(5)$ guarantees that the set $\pi _0(E)$ is $\kappa $-small. Moreover Corollary 3.2.5.5 shows that every nonempty fiber of the map $\pi _0(E_ b) \rightarrow \pi _0(E)$ is equipped with a transitive action of the fundamental group $\pi _{1}(B,b)$, which is also $\kappa $-small. Since $\kappa $ is regular, it follows that the set $\pi _0(E_ b)$ is also $\kappa $-small, as desired. $\square$