Kerodon

Proof. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{E}}$, and set $\overline{X} = U(X)$ and $\overline{Y} = U(Y)$. We wish to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is essentially $\kappa $-small. By virtue of Proposition 4.6.1.20, the functor $U$ induces a Kan fibration $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$. Our assumption that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small guarantees that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$ is essentially $\kappa $-small. By virtue of Corollary 5.4.7.2, it will suffice to show that for every morphism $\overline{e}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\overline{e}} = \{ \overline{e} \} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} ) } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is essentially $\kappa $-small. Since $U$ is a cocartesian fibration, we can lift $\overline{e}$ to a $U$-cocartesian morphism $e: X \rightarrow Y'$ of $\operatorname{\mathcal{E}}$. Proposition 5.1.3.11 then supplies a homotopy equivalence of $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\overline{e}}$ with the mapping space $\operatorname{Hom}_{ \operatorname{\mathcal{E}}_{ \overline{Y} } }( Y', Y )$ which is essentially $\kappa $-small by virtue of our assumption that $\operatorname{\mathcal{E}}_{ \overline{Y} }$ is locally $\kappa $-small. $\square$

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

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

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 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. $\square$

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 5.4.8.8 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 5.4.5.10. $\square$

Proof. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$, where $\operatorname{\mathcal{C}}'$ is an $\infty $-category. Using Proposition 5.7.7.2, we can write $U$ as the pullback of a cocartesian fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$. Proposition 5.3.6.1 then guarantees that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}'$ is a categorical equivalence of simplicial sets. Since the inclusion $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$ is bijective on vertices, every fiber of $U'$ can also be regarded as a fiber of $U$. We can therefore replace $U$ by $U'$, and thereby reduce to proving Corollary 5.4.8.11 in the special case where $U$ is a cocartesian fibration of $\infty $-categories.

Note that $U$ is an isofibration (Proposition 5.1.4.8). Consequently, for each object $C \in \operatorname{\mathcal{C}}$, the diagram

is a categorical pullback diagram of simplicial sets (Corollary 4.5.2.21). The implication $(1) \Rightarrow (2)$ now follows from Corollary 5.4.5.16 (and does not require the assumption that $\kappa $ is regular). To prove the reverse implication, we first note that the $\infty $-category $\operatorname{\mathcal{C}}$ and each fiber $\operatorname{\mathcal{E}}_{C}$ are locally $\kappa $-small (Example 5.4.8.4). Applying Proposition 5.4.8.7, we see that $\operatorname{\mathcal{E}}$ is locally $\kappa $-small. It will therefore suffice to show that the set of isomorphism classes $\pi _0( \operatorname{\mathcal{E}}^{\simeq } )$ is $\kappa $-small (Proposition 5.4.8.8). The functor $U$ induces a map $\theta : \pi _0( \operatorname{\mathcal{E}}^{\simeq } ) \rightarrow \pi _0( \operatorname{\mathcal{C}}^{\simeq } )$, whose target is $\kappa $-small. Invoking the regularity of $\kappa $, we are reduced to showing that for every element $[C] \in \pi _0( \operatorname{\mathcal{C}}^{\simeq } )$, the inverse image $\theta ^{-1} \{ [C] \} $ is a $\kappa $-small set. Let us identify $[C]$ with the isomorphism class of an object $C \in \operatorname{\mathcal{C}}$. Then there is a surjective map $\pi _0( \operatorname{\mathcal{E}}_{C}^{\simeq } ) \twoheadrightarrow \theta ^{-1} \{ [C] \} $. Since the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ is essentially $\kappa $-small, the set $\pi _0( \operatorname{\mathcal{E}}_{C}^{\simeq } )$ is $\kappa $-small, so that the quotient $\theta ^{-1} \{ [C] \} $ is also $\kappa $-small (Remark 5.4.3.4). $\square$