Kerodon

Proof. Since $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, we can choose an object $C \in \operatorname{\mathcal{C}}$ and a morphism $\overline{\beta }: (U \circ e) \rightarrow \underline{C}$ in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Using Theorem 5.2.1.1, we can lift $\overline{\beta }$ to a ($U$-cocartesian) natural transformation $\beta : e \rightarrow e'$, where $e'$ is a diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$. $\square$

Proof. Assume that $\operatorname{\mathcal{E}}$ is weakly contractible; we will show that $\operatorname{\mathcal{E}}$ is filtered (the converse follows from Proposition 9.1.1.13). Suppose that we are given a diagram $e: K \rightarrow \operatorname{\mathcal{E}}$, where $K$ is a finite simplicial set. We wish to show that there exists a natural transformation from $e$ to a constant diagram. Choose a monomorphism $K \hookrightarrow L$, where $L$ is another finite simplicial set which is weakly contractible (for example, we can take $L = K^{\triangleright }$). Let $\operatorname{Ex}^{\infty }( \operatorname{\mathcal{E}})$ be the simplicial set given by Construction 3.3.6.1, so that $\operatorname{Ex}^{\infty }( \operatorname{\mathcal{E}})$ is a Kan complex (Proposition 3.3.6.9). Let $\rho ^{\infty }: \operatorname{\mathcal{E}}\rightarrow \operatorname{Ex}^{\infty }( \operatorname{\mathcal{E}})$ be the weak homotopy equivalence of Proposition 3.3.6.7. Since $\operatorname{\mathcal{E}}$ is weakly contractible, the Kan complex $\operatorname{Ex}^{\infty }(\operatorname{\mathcal{E}})$ is contractible. It follows that the composite map $K \xrightarrow {e} \operatorname{\mathcal{E}}\xrightarrow { \rho ^{\infty } } \operatorname{Ex}^{\infty }( \operatorname{\mathcal{E}})$ can be extended to a map $e^{+}: L \rightarrow \operatorname{Ex}^{\infty }( \operatorname{\mathcal{E}})$. Since the simplicial set $L$ is finite, the morphism $\overline{e}$ factors through $\operatorname{Ex}^{m}( \operatorname{\mathcal{E}})$ for some $m \gg 0$ (see Proposition 3.6.1.9), and therefore corresponds to a diagram $f: \operatorname{Sd}^{m}(L) \rightarrow \operatorname{\mathcal{E}}$ such that $f|_{ \operatorname{Sd}^{m}(K) }$ is given by the composition $\operatorname{Sd}^{m}(K) \twoheadrightarrow K \xrightarrow {e} \operatorname{\mathcal{E}}$. Note that the quotient map $\operatorname{Sd}^{m}(K) \twoheadrightarrow K$ is universally localizing (Proposition 6.3.7.2) and therefore right cofinal (Corollary 7.2.1.11). We may therefore replace $K$ by $\operatorname{Sd}^{m}(K)$ and $L$ by $\operatorname{Sd}^{m}(L)$, and thereby reduce to the case where $m = 0$: that is, the diagram $e$ admits an extension $f: L \rightarrow \operatorname{\mathcal{E}}$. Using Lemma 9.1.3.1, we can choose a natural transformation $\beta : f \rightarrow f'$, where $f': L \rightarrow \operatorname{\mathcal{E}}$ factors through the Kan complex $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ for some object $C \in \operatorname{\mathcal{C}}$. Since $L$ is weakly contractible, the morphism $f'$ is nullhomotopic (when regarded as a diagram in the Kan complex $\operatorname{\mathcal{E}}_{C}$). We may therefore assume without loss of generality that $f' = \underline{X}$ is the constant diagram associated to some object $X \in \operatorname{\mathcal{E}}_{C}$. Restricting to the simplicial subset $K \subseteq L$, we obtain a natural transformation from $e = f|_{K}$ to the constant diagram $f'|_{K}$. $\square$

Proof. By virtue of Proposition 7.4.3.1, the colimit $\varinjlim ( \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )$ is contractible if and only if the $\infty $-category $\operatorname{\mathcal{E}}$ is weakly contractible. The desired result is therefore a reformulation of Theorem 9.1.3.2. $\square$

Proof. Without loss of generality, we may assume that $\kappa $ is regular (and therefore of cofinality $> \aleph _0$). Let $Y_0 \subseteq X$ be the image of $e$. We will construct $Y$ as the union of an increasing sequence of $\kappa $-small simplicial subsets

Assume that $Y_ n$ has been defined for some $n \geq 0$. Since $X$ is weakly contractible, the Kan complex $\operatorname{Ex}^{\infty }(X)$ is contractible. It follows that the inclusion $\operatorname{Ex}^{\infty }(Y_ n) \hookrightarrow \operatorname{Ex}^{\infty }(X)$ is nullhomotopic: that is, it admits an extension $f_ n: \operatorname{Ex}^{\infty }(Y_ n)^{\triangleright } \rightarrow \operatorname{Ex}^{\infty }(X)$. Since $\operatorname{Ex}^{\infty }(Y_ n)^{\triangleright }$ is $\kappa $-small (Corollary 4.7.4.16), we can choose a $\kappa $-small simplicial $Y_{n+1} \subseteq X$ containing $Y_{n}$ such that $f_ n$ factors through $\operatorname{Ex}^{\infty }( Y_{n+1} )$. To complete the proof, it will suffice to show that the union $Y = \bigcup _{n} Y_{n}$ is weakly contractible, or equivalently that the Kan complex $\operatorname{Ex}^{\infty }(Y)$ is contractible. This follows from Corollary 3.2.4.15, since $\operatorname{Ex}^{\infty }(Y)$ is the colimit of a diagram of nullhomotopic maps

Proof of Variant 9.1.3.6. Assume that $\operatorname{\mathcal{E}}$ is weakly contractible; we will show that it is $\kappa $-filtered (the converse follows from Proposition 9.1.1.13). If $\kappa = \aleph _0$, this follows from Theorem 9.1.3.2. We may therefore assume without loss of generality that $\kappa $ is uncountable. Fix a diagram $e: K \rightarrow \operatorname{\mathcal{E}}$, where $K$ is a $\kappa $-small simplicial set; we wish to show that there exists a natural transformation from $e$ to a constant diagram. Using Lemma 9.1.3.5, we can reduce to the case where $K$ is weakly contractible. Using Lemma 9.1.3.1, we can reduce further to the situation where $e$ factors through the Kan complex $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ for some object $C \in \operatorname{\mathcal{C}}$. In this case, our assumption that $K$ is weakly contractible guarantees that $e$ is nullhomotopic (when regarded as a diagram $\operatorname{\mathcal{E}}_{C}$), and therefore isomorphic to a constant diagram (when regarded as a diagram in $\operatorname{\mathcal{E}}$). $\square$

Proof of Proposition 9.1.3.8. The implications $(1) \Rightarrow (2)$ and $(3) \Rightarrow (2)$ are immediate from the definitions, and the implication $(2) \Rightarrow (1)$ follows from Lemma 9.1.3.1. We will complete the proof by showing that $(2) \Rightarrow (3)$. Let $C \in \operatorname{\mathcal{C}}$ be an object and let $e: K \rightarrow \operatorname{\mathcal{E}}_{C}$ be a diagram, where $K$ is a $\kappa $-small simplicial set. It follows from $(2)$ that there exists an object $X \in \operatorname{\mathcal{E}}$ and a natural transformation $\beta : e \rightarrow \underline{X}$ in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{E}})$. Set $C' = U(X)$ and form a pushout diagram of simplicial sets

By construction, $U(\beta )$ corresponds to a diagram in $\operatorname{\mathcal{C}}$ which factors as a composition

Since $L$ is $\kappa $-small and $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, the diagram $s$ admits an extension $\overline{s}: L^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Let $D \in \operatorname{\mathcal{C}}$ be the image under $\overline{s}$ of the cone point of $L^{\triangleright }$. Restricting $\overline{s}$ to the edges $\{ C\} ^{\triangleright } \subseteq L^{\triangleright }$ and $\{ C'\} ^{\triangleright } \subseteq L^{\triangleright }$, we obtain a pair of morphisms $f: C \rightarrow D$ and $g: C' \rightarrow D$. Note that $q$ extends to a morphism $\overline{q}: \Delta ^2 \times K \rightarrow L^{\triangleright }$, carrying $\{ 2\} \times K$ to the cone point of $L^{\triangleright }$. Let us identify the composition $\overline{s} \circ \overline{q}$ with a $2$-simplex $\sigma $ of the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, which we depict as a diagram

Since $U$ is a cocartesian fibration, we can write $g = U( \widetilde{g} )$ for some morphism $\widetilde{g}: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$. Let $\gamma : \underline{X} \rightarrow \underline{Y}$ denote the image of $\widetilde{g}$ under the diagonal map $\operatorname{\mathcal{E}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{E}})$. Using Corollary 4.1.4.2, we can lift $\sigma $ to a $2$-simplex

of the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{E}})$. By construction, we have $U( \alpha ) = \underline{f}$, so the morphism $f$ satisfies condition $(\ast )$. $\square$