Kerodon

Proof. The implication $(1) \Rightarrow (2)$ follows from Example 4.7.8.4. 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

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 4.7.7.2 (and Remark 4.7.5.8), 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 4.7.7.1 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

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

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 4.7.5.17). 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 4.7.5.14, 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 4.7.7.1. 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 4.7.7.1. Using Proposition 4.7.6.15, 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 4.7.4.10, 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 4.7.3.4 and Proposition 4.7.3.5), 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 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 3.2.6.5 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. $\square$

Proof. Let $F,F': K \rightarrow \operatorname{\mathcal{C}}$ be diagrams; we wish to show that the morphism space $\operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}(F,F')$ is essentially $\lambda $-small. Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the essential images of $F$ and $F'$. Proposition 4.7.8.7 guarantees that $\operatorname{\mathcal{C}}_0$ is essentially $\lambda $-small. It will therefore suffice to show that $\operatorname{Fun}(K, \operatorname{\mathcal{C}}_0)$ is locally $\lambda $-small, which follows immediately from Remark 4.7.5.11. $\square$

Proof. We will show that the slice $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ is locally $\lambda $-small; the analogous assertion for $\operatorname{\mathcal{C}}_{f/}$ follows by a similar argument. Fix a pair of objects $X,Y \in \operatorname{\mathcal{C}}_{/f}$; we wish to show that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{/f}}( X, Y)$ is essentially $\lambda $-small. Let $\operatorname{\mathcal{C}}'$ be the smallest full subcategory of $\operatorname{\mathcal{C}}$ which contains $f(K)$ together with the images of $X$ and $Y$. Replacing $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$, we can reduce to the case where $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. In this case, the $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ is essentially $\lambda $-small (Corollary 4.7.5.18), so the desired result follows from Example 4.7.8.4. $\square$