Kerodon

Proof. Let $J$ be a set of cardinality $\kappa $ and let $\operatorname{\mathcal{J}}$ be the corresponding indiscrete category (that is, the category having object set $\operatorname{Ob}(\operatorname{\mathcal{J}}) = J$ and $\operatorname{Hom}_{\operatorname{\mathcal{J}}}(j,j') = \ast $ for every pair of elements $j,j' \in J$). The nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{J}})$ is a contractible Kan complex. Setting $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{N}_{\bullet }(\operatorname{\mathcal{J}}) \times \operatorname{\mathcal{C}}$, it follows that the projection map $\pi : \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration. We will complete the proof by showing that $\widetilde{\operatorname{\mathcal{C}}}$ satisfies condition $(\ast )$. Let $K$ be a $\kappa $-small simplicial subset of $\widetilde{\operatorname{\mathcal{C}}}$, so that the inclusion map $K \hookrightarrow \widetilde{\operatorname{\mathcal{C}}}$ can be identified with a pair of morphisms

Since $J$ has cardinality $\kappa $, we can choose an element $j \in J$ which is not of the form $f(x)$ for any vertex $x \in K$. It follows that $f$ admits a unique extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{J}})$ which carries the cone point of $K^{\triangleright }$ to the element $j \in J$. Our assumption that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered guarantees that $g$ admits an extension $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. We complete the proof by observing that the pair $(\overline{f}, \overline{g} )$ determines a monomorphism of simplicial sets $K^{\triangleright } \rightarrow \widetilde{\operatorname{\mathcal{C}}}$. $\square$

Proof. Fix a regular cardinal $\lambda $ such that $\operatorname{\mathcal{C}}$ is $\lambda $-small. If $\kappa \trianglelefteq \lambda $, then we can choose a $\lambda $-small collection $\{ K_{\alpha } \} _{\alpha \in A}$ of $\kappa $-small simplicial subsets of $\operatorname{\mathcal{C}}$, such that every $\kappa $-small simplicial subset of $\operatorname{\mathcal{C}}$ is contained in some $K_{\alpha }$ (Lemma 9.1.7.18). Since $\operatorname{\mathcal{C}}$ satisfies condition $(\ast )$, each $K_{\alpha }$ is contained in a larger simplicial subset $X_{\alpha } \subseteq \operatorname{\mathcal{C}}$ which is isomorphic to the cone $K_{\alpha }^{\triangleright }$. We regard $A$ as a partially ordered set, where $\alpha \leq \beta $ if $X_{\alpha }$ is contained in $X_{\beta }$. Then $(A, \leq )$ is automatically $\kappa $-directed (Remark 9.1.7.19), and the colimit $\varinjlim _{\alpha \in A} X_{\alpha }$ identifies with the union $\bigcup _{\alpha } X_{\alpha } = \operatorname{\mathcal{C}}$. $\square$

Proof of Proposition 9.1.8.5. Replacing $\operatorname{\mathcal{C}}$ by an equivalent $\infty $-category, we may assume that it is $\lambda $-small and satisfies condition $(\ast )$ of Lemma 9.1.8.2. Using Lemma 9.1.8.4, we can realize $\operatorname{\mathcal{C}}$ as the colimit of a diagram

where each $X_{\alpha }$ has the form $K_{\alpha }^{\triangleright }$ for some $\kappa $-small simplicial set $K_{\alpha }$. Using Proposition 4.1.3.2, we can choose a levelwise categorical equivalence $\mathscr {F} \rightarrow \mathscr {F}'$ for some functor

carrying each $\alpha \in A$ to an $\infty $-category $\operatorname{\mathcal{E}}_{\alpha }$. By construction, each of the $\infty $-categories $\operatorname{\mathcal{E}}_{\alpha }$ essentially $\lambda $-small, so that $\mathscr {F}'$ determines a diagram $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$. It follows from Corollary 9.1.6.3 that $\operatorname{\mathcal{C}}$ is a colimit of this diagram. $\square$

Proof. Using Proposition 9.1.8.5 (and Remark 9.1.8.6), we can choose a $\lambda $-small, $\kappa $-directed partially ordered set $(A, \leq )$ and a diagram

having colimit $\operatorname{\mathcal{C}}$, where each $\operatorname{\mathcal{E}}_{\alpha }$ has a final object. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(A)$ be a cocartesian fibration with covariant transport representation $\mathscr {F}$ (for example, we can take $\operatorname{\mathcal{E}}$ to be the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ of Definition 5.6.2.1) and let $W$ be the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. Since $\operatorname{\mathcal{C}}$ is a colimit of the diagram $\mathscr {F}$, there is a functor $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which exhibits $\operatorname{\mathcal{C}}$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$ (Proposition 7.4.5.20), and is therefore right cofinal (Proposition 7.2.1.10). Since each fiber of $U$ has a final object, the functor $U$ is a reflective localization (Corollary 7.1.5.22): that is, it has a (fully faithful) right adjoint $V: \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{E}}$. The functor $V$ is also right cofinal (Corollary 7.2.3.7), so the composition $(G \circ V): \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$ is right cofinal. $\square$

Proof. Combine Corollary 9.1.5.11 with Theorem 9.1.8.7. $\square$

Proof. Using Theorem 9.1.8.7, we can reduce to the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(A)$, where $(A, \leq )$ is a countable directed partially ordered set. In this case, we can identify a functor $F: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$ with a nondecreasing sequence $a_0 \leq a_1 \leq a_2 \leq \cdots $. To guarantee that $F$ is right cofinal, it suffices to choose a sequence with the property that each $a \in A$ satisfies $a \leq a_ n$ for $n \gg 0$ (see Theorem 7.2.3.1). $\square$

Proof. The equivalence $(2) \Leftrightarrow (3)$ follows from Remark 8.5.6.19 (and does not require the assumption that $\operatorname{\mathcal{C}}$ is $\kappa $-small). Since the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{Idem})$ is $\kappa $-filtered (Example 9.1.1.8), the implication $(3) \Rightarrow (1)$ is a special case of Corollary 9.1.5.11. We will complete the proof by showing that $(1)$ implies $(2)$. If condition $(1)$ is satisfied, then there exists an object $X \in \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \underline{X}$ in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. Fix an uncountable cardinal $\lambda $ such that $\operatorname{\mathcal{C}}$ is locally $\lambda $-small, let $\widehat{\operatorname{\mathcal{C}}}$ denote the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, and let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Since $H$ is dense (Theorem 8.4.2.1), every object $\mathscr {F} \in \widehat{\operatorname{\mathcal{C}}}$ can be recovered as the colimit of the diagram

In particular, if $\mathscr {F}$ is a final object of $\widehat{\operatorname{\mathcal{C}}}$, then it is a colimit of the diagram $H$. It follows that there exists a morphism $i: \mathscr {F} \rightarrow H(X)$ in the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ and a commutative diagram

in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \widehat{\operatorname{\mathcal{C}}} )$. Since $\mathscr {F}$ is a final object of $\widehat{\operatorname{\mathcal{C}}}$, the morphism $i$ automatically has a left homotopy inverse (given by any morphism $r: H(X) \rightarrow \mathscr {F}$). In particular, $\mathscr {F}$ is a retract of $H(X)$, and can therefore be viewed as a final object in the idempotent completion of $\operatorname{\mathcal{C}}$ (see Proposition 8.5.5.5). $\square$

Proof. This follows from Proposition 9.1.8.10 (applied in the special case $\kappa = \aleph _0$), since the $\infty $-category $\operatorname{\mathcal{C}}$ is automatically idempotent complete (Example 8.5.4.5). $\square$