Kerodon

Proof. For each $t \in T$, set $S_{\geq t} = \{ s \in S: t \leq f(s) \} $, which we regard as a linearly ordered subset of $S$. Using Theorem 7.2.3.1, we can rewrite conditions $(1)$ and $(2)$ as follows:

$(1')$

For each element $t \in T$, the linearly ordered set $S_{\geq t}$ is nonempty.

$(2')$

For each element $t \in T$, the linearly ordered set $S_{\geq t}$ has weakly contractible nerve.

The implication $(2') \Rightarrow (1')$ is immediate, and the reverse implication follows from Corollary 3.2.8.5. $\square$

Proof. Without loss of generality, we may assume that $f$ carries each edge of $A$ to an isomorphism in $\operatorname{\mathcal{C}}$. Under this assumption, we can restate $(1)$ and $(2)$ as follows:

$(1')$

For every vertex $a \in A$, the edge

\[ \Delta ^1 \simeq \{ a\} ^{\triangleleft } \hookrightarrow A^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]

is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

$(2')$

The morphism $\overline{f}$ is a colimit diaram.

Using Corollary 3.1.7.2, we can choose an anodyne morphism $i: A \hookrightarrow B$, where $B$ is a Kan complex. Note that $f$ can be regarded as a morphism from $A$ to the core $\operatorname{\mathcal{C}}^{\simeq }$, which is also a Kan complex (Corollary 4.4.3.11). We can therefore extend $f$ to a morphism of Kan complexes $g: B \rightarrow \operatorname{\mathcal{C}}^{\simeq }$. Moreover, the morphism $i$ is right cofinal (Proposition 7.2.1.5) and therefore right anodyne (Proposition 7.2.1.3). It follows that the induced map $B \coprod _{A} A^{\triangleright } \hookrightarrow B^{\triangleright }$ is inner anodyne (Example 4.3.6.5), so that we can choose a functor $\overline{g}: B^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{g}|_{B} = g$ and $\overline{g}|_{A^{\triangleright }} = \overline{f}$.

It follows from Corollary 7.2.2.3 $\overline{f}$ is a colimit diagram if and only if $\overline{g}$ is a colimit diagram. Since $A$ is weakly contractible, the Kan complex $B$ is contractible. In particular, every vertex $a \in A$ can be regarded as a final object of $B$. The equivalence of $(1')$ and $(2')$ now follows from Corollary 7.2.2.6. $\square$

Proof. Combine Corollary 7.2.2.6 with Proposition 7.1.5.14. $\square$

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Suppose that $F$ admits a right $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. For every object $X \in \operatorname{\mathcal{D}}$, Corollary 6.2.4.2 guarantees that the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X}$ has a final object. In particular, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X}$ is weakly contractible (Corollary 4.6.6.25). Allowing $X$ to vary and applying Theorem 7.2.3.1, we conclude that $F$ is left cofinal. $\square$

Proof. By virtue of Remark 7.2.3.2, condition $(1)$ is equivalent to the requirement that for every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the oriented fiber product $\{ f\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ is weakly contractible. The equivalence of $(1)$ and $(2)$ now follows from Theorem 4.6.4.17. $\square$

Proof. By virtue of Theorem 7.2.3.1, it will suffice to show that for every integer $n \geq 0$, the category $\operatorname{\mathcal{C}}= \operatorname{{\bf \Delta }}_{\operatorname{inj}} \times _{\operatorname{{\bf \Delta }}} \operatorname{{\bf \Delta }}_{ / [n] }$ has weakly contractible nerve. Let $C_0 \in \operatorname{\mathcal{C}}$ denote the object corresponding to the inclusion map $[0] \simeq \{ n\} \hookrightarrow [n]$. For every object $C \in \operatorname{\mathcal{C}}$, given by a nondecreasing function $\alpha : [m] \rightarrow [n]$, we let $F(C) \in \operatorname{\mathcal{C}}$ denote the object given by the nondecreasing function $\alpha ^{+}: [m+1] \rightarrow [n]$ given by the formula

Note that we have canonical maps $C \xrightarrow {\beta _{-}} F(C) \xleftarrow {\beta _+} C_0$, given by the inclusions

These morphisms depend functorially on $C$, and therefore furnish natural transformations of functors $\operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow F \leftarrow \underline{C}_0$, where $\underline{C}_0: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ denotes the constant functor taking the value $C_0$. It follows that the identity morphism of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is homotopic to the constant morphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \twoheadrightarrow \{ C_0 \} \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, so that the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is contractible (and, in particular, it is weakly contractible). $\square$

Proof. Without loss of generality, we may assume that $f$ is the inclusion map $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for some $0 < i \leq n$. Using Remark 5.3.2.3, we can reduce to the case where $\operatorname{\mathcal{C}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ and $g$ is the identity map. In this case, Remark 5.3.2.12 supplies a pushout diagram of simplicial sets

It will therefore suffice to show that the left vertical map is right anodyne, which follows from Proposition 4.2.5.3. $\square$

Proof. By virtue of Corollary 7.2.1.15, it will suffice to prove Proposition 7.2.3.13 in the special case where $\overline{F}$ is right anodyne. Let $S$ be the collection of all morphisms of simplicial sets $\overline{F}: \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}$ having the property that, for every cocartesian fibration $\pi : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the induced map $F: \operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is right anodyne. We wish to show show that every right anodyne morphism belongs to $S$. It follows immediately from the definitions that $S$ is weakly saturated, in the sense of Definition 1.4.4.15. It will therefore suffice to show that $S$ contains every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $0 < i \leq n$. In other words, we are reduced to proving Proposition 7.2.3.13 in the special case where $\operatorname{\mathcal{D}}= \Delta ^ n$ is a standard simplex and $\overline{F}$ is the inclusion of the horn $\Lambda ^{n}_{i} \subseteq \Delta ^ n$.

Applying Corollary 5.3.4.9, we deduce that there exists a diagram of $\infty $-categories $\mathscr {G}: [n] \rightarrow \operatorname{QCat}$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \rightarrow \operatorname{\mathcal{C}}$ for the cocartesian fibration $\pi $. We then have a commutative diagram of simplicial sets

where $F'$ is right anodyne (Lemma 7.2.3.11) and therefore right cofinal (Proposition 7.2.1.3). Lemma 5.3.6.4 guarantees that horizontal maps are categorical equivalences, so that $F$ is also right cofinal (Corollary 7.2.1.22). $\square$

Proof. Combine Propositions 7.2.3.13 and 7.2.1.3. $\square$

Proof of Theorem 7.2.3.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. We will show that $F$ is right cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is weakly contractible; the analogous characterization of left cofinal morphisms follows by a similar argument.

Suppose first that $F$ is right cofinal. For every object $X \in \operatorname{\mathcal{D}}$, the projection map $\operatorname{\mathcal{D}}_{X/} \rightarrow \operatorname{\mathcal{D}}$ is a left fibration (Proposition 4.3.6.1), and therefore a cocartesian fibration (Proposition 5.1.4.14). Applying Proposition 7.2.3.13, we conclude that the projection map $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/} \rightarrow \operatorname{\mathcal{D}}_{X/}$ is also right cofinal. In particular, it is a weak homotopy equivalence (Proposition 7.2.1.5). Since the $\infty $-category $\operatorname{\mathcal{D}}_{X/}$ has an initial object (Proposition 4.6.6.22), it is weakly contractible, so that the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is also weakly contractible.

We now prove the converse. Assume that, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is weakly contractible. We wish to show that $F$ is right cofinal. Using Proposition 4.1.3.2, we can factor $F$ as a composition

where $F'$ is inner anodyne and $F''$ is an inner fibration. Since $F'$ is right cofinal (Proposition 7.2.1.3), it will suffice to show that $F''$ is right cofinal (Proposition 7.2.1.6). For every object $X \in \operatorname{\mathcal{D}}$, Proposition 5.3.6.1 guarantees that the induced map $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/} \hookrightarrow \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is a categorical equivalence. In particular, it is a weak homotopy equivalence (Remark 4.5.3.4), so that $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is also weakly contractible. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$ and thereby reduce to the case where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an inner fibration, so that $\operatorname{\mathcal{C}}$ is also an $\infty $-category (Remark 4.1.1.9).

Let $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$ denote the functors given by evaluation at the vertices $0,1 \in \operatorname{\mathcal{D}}$, and let $\delta : \operatorname{\mathcal{D}}\hookrightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})$ be the diagonal map. Note that there is a unique natural transformation from $\operatorname{id}_{ \Delta ^1 }$ to the constant map $\Delta ^1 \twoheadrightarrow \{ 1\} \hookrightarrow \Delta ^1$, which induces a natural transformation $h: \operatorname{id}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}}) } \rightarrow \delta \circ \operatorname{ev}_{1}$. Let $\operatorname{\mathcal{M}}$ denote the oriented fiber product $\operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}= \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{D}}) } \operatorname{Fun}(\{ 1\} , \operatorname{\mathcal{C}})$ of Construction 4.6.4.1, so that $\operatorname{ev}_{0}$ and $\operatorname{ev}_{1}$ lift to functors

the diagonal map $\delta $ lifts to a functor $\widetilde{\delta }: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{M}}$, and $h$ lifts to a natural transformation $\widetilde{h}: \operatorname{id}_{\operatorname{\mathcal{M}}} \rightarrow \widetilde{\delta } \circ \widetilde{\operatorname{ev}}_1$. Note that $\widetilde{h}$ can be identified with a morphism of simplicial sets $\Delta ^1 \times \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{M}}$ which fits into a commutative diagram

where the horizontal compositions are the identity. It follows that $\widetilde{\delta }$ is a retract of $\iota $. Since $\iota $ is right anodyne (Proposition 4.2.5.3), $\widetilde{\delta }$ is also right anodyne, and therefore right cofinal (Proposition 7.2.1.3).

The functor $\widetilde{\operatorname{ev}}_0: \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{D}}$ is a cartesian fibration (Corollary 5.3.7.3). Moreover, for each object $X \in \operatorname{\mathcal{D}}$, the fiber $\widetilde{\operatorname{ev}}_{0}^{-1} \{ X\} \simeq \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is equivalent to the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ (Example 5.1.7.7), and is therefore weakly contractible. Applying Corollary 6.3.5.3, we deduce that the functor $\widetilde{\operatorname{ev}}_1$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$, and is therefore right cofinal (Proposition 7.2.1.10). We now observe that the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ factors as a composition

and is therefore also right cofinal (Proposition 7.2.1.6). $\square$

Proof. We first reduce to the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $\iota : \operatorname{\mathcal{C}}\hookrightarrow \overline{\operatorname{\mathcal{C}}}$, where $\overline{\operatorname{\mathcal{C}}}$ is an $\infty $-category. Using Proposition 5.7.7.2, we can extend $U$ and $U'$ to cocartesian fibrations of $\infty $-categories $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \overline{\operatorname{\mathcal{C}}}$ and $\overline{U}': \overline{\operatorname{\mathcal{E}}}' \rightarrow \overline{\operatorname{\mathcal{C}}}$. Then the inclusion maps $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$ and $\operatorname{\mathcal{E}}' \hookrightarrow \overline{\operatorname{\mathcal{E}}}'$ are categorical equivalences (Lemma 5.3.6.5). Since $\overline{U}$ is an isofibration (Proposition 5.1.4.8), we can extend $F$ to a functor $\overline{F}: \overline{\operatorname{\mathcal{E}}}' \rightarrow \overline{\operatorname{\mathcal{E}}}$ satisfying $\overline{U} \circ \overline{F} = \overline{U}'$ (Proposition 4.5.5.1). It follows from Remark 5.3.1.12 that the functor $\widetilde{F}$ carries $\overline{U}'$-cartesian morphisms of $\overline{\operatorname{\mathcal{E}}}'$ to $\overline{U}$-cartesian morphisms of $\overline{\operatorname{\mathcal{E}}}$. Moreover, the morphism $F$ is right cofinal if and only if $\overline{F}$ is right cofinal (Corollary 7.2.1.22). Consequently, we can replace $\operatorname{\mathcal{C}}$ by $\overline{\operatorname{\mathcal{C}}}$ and thereby reduce to proving Corollary 7.2.3.16 in the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category.

Fix an object $X \in \operatorname{\mathcal{E}}$, let $C = U(X)$ denote its image in $\operatorname{\mathcal{C}}$, and write $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{E}}'_{C}$ denote the fibers $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$, respectively. We will prove that the following conditions are equivalent:

$(1_ X)$

The $\infty $-category $\operatorname{\mathcal{E}}' \times _{ \operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_{ X/ }$ is weakly contractible.

$(2_ X)$

The $\infty $-category $\operatorname{\mathcal{E}}'_{C} \times _{ \operatorname{\mathcal{E}}_{C} } (\operatorname{\mathcal{E}}_{C})_{X/}$ is weakly contractible.

Corollary 7.2.3.16 will then follow by allowing the object $X$ to vary and applying the criterion of Theorem 7.2.3.1.

To complete the proof, it will suffice to show that the inclusion map

is a weak homotopy equivalence. In fact, we will show that it is left anodyne. Unwinding the definitions, we have a pullback diagram

where the right vertical map is a cartesian fibration (Corollary 5.1.4.21). By virtue of Proposition 7.2.3.13, we are reduced to showing that the inclusion map $\{ \operatorname{id}_{C} \} \hookrightarrow \operatorname{\mathcal{C}}_{C/}$ is left anodyne, or equivalently that $\{ \operatorname{id}_{C} \} $ is an initial object of the $\infty $-category $\operatorname{\mathcal{C}}_{C/}$ (Corollary 4.6.6.23). This is a special case of Proposition 4.6.6.22. $\square$