Kerodon

Proof. By virtue of Remark 9.1.1.10, it will suffice to show that for every $\kappa $-small simplicial set $L$ and every morphism $g: L \rightarrow \operatorname{\mathcal{C}}_{f/}$, the $\infty $-category $(\operatorname{\mathcal{C}}_{f/} )_{g/}$ is nonempty. Unwinding the definitions, we can identify $g$ with a morphism of simplicial sets $\overline{f}: K \star L \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{f}|_{K} = f$. This identification supplies an isomorphism $(\operatorname{\mathcal{C}}_{f/} )_{g/} \simeq \operatorname{\mathcal{C}}_{ \overline{f} / }$. We are therefore reduced to showing that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ \overline{f} / }$ is nonempty. This follows from Remark 9.1.1.10, since the simplicial set $K \star L$ is $\kappa $-small (Corollary 4.7.4.13). $\square$

Proof. By virtue of Proposition 3.1.7.1, there exists a functor $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ and a natural transformation $u: \operatorname{id}_{\operatorname{Set_{\Delta }}} \rightarrow Q$ with the following properties:

• The functor $Q$ commutes with filtered colimits.

• For every simplicial set $X$, the simplicial set $Q(X)$ is a Kan complex.

• For every simplicial set $X$, the morphism $u_{X}: X \rightarrow Q(X)$ is a weak homotopy equivalence.

To show that $\operatorname{\mathcal{C}}$ is weakly contractible, it will suffice to show that the Kan complex $Q(\operatorname{\mathcal{C}})$ is contractible. Note that $\operatorname{\mathcal{C}}$ is nonempty, so that $Q(\operatorname{\mathcal{C}})$ is also nonempty. It will therefore suffice to show that for every integer $n \geq 0$, every morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^ n \rightarrow Q(\operatorname{\mathcal{C}})$ is nullhomotopic (see Variant 3.2.4.12). Since the simplicial set $\operatorname{\partial \Delta }^ n$ is finite and the functor $Q$ commutes with filtered colimits, the morphism $\sigma $ factors as a composition $\operatorname{\partial \Delta }^ n \rightarrow Q(K) \xrightarrow {Q(\iota )} Q(\operatorname{\mathcal{C}})$, where $K$ is a finite simplicial subset of $\operatorname{\mathcal{C}}$ and $\iota : K \hookrightarrow \operatorname{\mathcal{C}}$ denotes the inclusion map. We will complete the proof by showing that $Q(\iota )$ is nullhomotopic. Since $u_{K}: K \rightarrow Q(K)$ is a weak homotopy equivalence, this is equivalent to assertion that the composite morphism $Q(\iota ) \circ u_{K} = u_{\operatorname{\mathcal{C}}} \circ \iota $ is nullhomotopic. This is clear: our assumption that $\operatorname{\mathcal{C}}$ is filtered guarantees that there exists a natural transformation from $\iota $ to a constant diagram $K \rightarrow \operatorname{\mathcal{C}}$ (Remark 9.1.1.10). $\square$

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 9.1.1.12, the implication $(2) \Rightarrow (3)$ from Remark 9.1.1.8, the implication $(3) \Rightarrow (4)$ Proposition 9.1.1.13, and the implication $(4) \Rightarrow (1)$ is immediate from the definitions (Remark 9.1.1.10). The equivalence $(4) \Leftrightarrow (5)$ is a special case of Corollary 7.2.3.9. $\square$

Proof. By virtue of Proposition 9.1.1.14, it will suffice to show that for every ($\kappa $-small) simplicial set $K$, the diagonal map $\delta _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is right cofinal if and only if the diagonal map $\delta _{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{D}})$ is right cofinal. This follows by applying Corollary 7.2.1.22 to the commutative diagram of $\infty $-categories

Proof. If $\operatorname{\mathcal{C}}$ is a contractible Kan complex, then there exists a categorical equivalence $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$, so that $\operatorname{\mathcal{C}}$ is filtered by virtue of Corollary 9.1.1.15. The converse is a special case of Proposition 9.1.1.13. $\square$