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7.2.3 Quillen's Theorem A for $\infty $-Categories

The following result provides a concrete criterion for establishing the cofinality of a functor between $\infty $-categories.

Theorem 7.2.3.1 (Joyal). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. Then:

$(1)$

The morphism $F$ is left 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.

$(2)$

The morphism $F$ 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.

Remark 7.2.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. For every object $X \in \operatorname{\mathcal{D}}$, the slice and coslice diagonal morphisms of Construction 4.6.4.13 induce categorical equivalences

\[ \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \quad \quad \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/} \hookrightarrow \{ X\} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{C}} \]

(Example 5.1.6.7). We can therefore reformulate Theorem 7.2.3.1 as follows:

$(1')$

The morphism $F$ is left cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} $ is weakly contractible.

$(2')$

The morphism $F$ is right cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ is weakly contractible.

Example 7.2.3.3 (Quillen's Theorem A). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Suppose that, for every object $X \in \operatorname{\mathcal{D}}$, the category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ has weakly contractible nerve. Applying Theorem 7.2.3.1, we deduce that the induced morphism of simplicial sets $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is right cofinal. In particular, it is a weak homotopy equivalence (Proposition 7.2.1.5). This recovers a classical result of Quillen (see [MR2655184]).

Corollary 7.2.3.4. Let $(S, \leq )$ and $(T, \leq )$ be linearly ordered sets, and let $f: S \rightarrow T$ be a nondecreasing function. The following conditions are equivalent:

$(1)$

The function $f: S \rightarrow T$ is cofinal in the sense of Definition 5.4.1.26. That is, for every element $t \in T$, there exists an element $s \in S$ satisfying $t \leq f(s)$.

$(2)$

The induced morphism of simplicial sets $\operatorname{N}_{\bullet }(S) \rightarrow \operatorname{N}_{\bullet }(T)$ is right cofinal, in the sense of Definition 7.2.1.1.

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. $\square$

Corollary 7.2.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\overline{f}: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a diagram, where $A$ is a weakly contractible simplicial set. The following conditions are equivalent:

$(1)$

The diagram $\overline{f}$ carries each edge of $A^{\triangleleft }$ to an isomorphism in $\operatorname{\mathcal{C}}$.

$(2)$

The restriction $f = \overline{f}|_{A}$ carries each edge of $A$ to an isomorphism in $\operatorname{\mathcal{C}}$, and $\overline{f}$ is a limit diagram.

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. $\square$

Corollary 7.2.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories. Then:

  • If $\operatorname{\mathcal{C}}$ has an initial object $Y$ and $\overline{F}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a functor which carries $\{ Y\} ^{\triangleleft } \simeq \Delta ^1$ to an isomorphism in $\operatorname{\mathcal{D}}$, then $\overline{F}$ is a $U$-limit diagram.

  • If $\operatorname{\mathcal{C}}$ has an initial object $Y$ and $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a functor which carries $\{ Y\} ^{\triangleright } \simeq \Delta ^1$ to an isomorphism in $\operatorname{\mathcal{D}}$, then $\overline{F}$ is a $U$-limit diagram.

Corollary 7.2.3.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:

$(1)$

If $F$ is a left adjoint, then it is left cofinal.

$(2)$

If $F$ is a right adjoint, then it is right cofinal.

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 4.6.6.26). Allowing $X$ to vary and applying Theorem 7.2.3.1, we conclude that $F$ is left cofinal. $\square$

Example 7.2.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is a reflective subcategory (Definition 6.2.2.1), then the inclusion map $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is right cofinal (this is a special case of Corollary 7.2.3.7, since Proposition 6.2.2.8 guarantees that $\iota $ has a left adjoint). Similarly, if $\operatorname{\mathcal{C}}_0$ is a coreflective subcategory of $\operatorname{\mathcal{C}}$, then the inclusion $\iota $ is left cofinal.

Corollary 7.2.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. The following conditions are equivalent:

$(1)$

The diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is right cofinal.

$(2)$

For every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is weakly contractible.

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. $\square$

Corollary 7.2.3.10. Let $\operatorname{{\bf \Delta }}$ denote the simplex category (Definition 1.1.1.2), and let $\operatorname{{\bf \Delta }}_{\operatorname{inj}} \subset \operatorname{{\bf \Delta }}$ denote the non-full subcategory whose morphisms are strictly increasing functions $[m] \hookrightarrow [n]$ (Variant 1.1.1.6). Then the inclusion of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})$ is left cofinal.

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

\[ \alpha ^{+}(i) = \begin{cases} \alpha (i) & \text{ if } 0 \leq i \leq m \\ n & \text{ if } i = m+1. \end{cases} \]

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

\[ \{ 0 < 1 < \cdots < m \} \hookrightarrow \{ 0 < 1 < \cdots < m+1 \} \hookleftarrow \{ m+1 \} . \]

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). $\square$

Our proof of Theorem 7.2.3.1 will require some preliminaries.

Lemma 7.2.3.11. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$. Suppose we are given morphisms of simplicial sets $A \xrightarrow {f} B \xrightarrow {g} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $f$ is right anodyne. Then the induced map $A \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow B \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is right anodyne.

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

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \times \mathscr {F}(0) \ar [r] \ar [d] & \Lambda ^{n}_{i} \times _{ \Delta ^ n} \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [d] \\ \Delta ^ n \times \mathscr {F}(0) \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ). } \]

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

Example 7.2.3.12. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and suppose that the category $\operatorname{\mathcal{C}}$ contains a final object $C$. Combining Lemma 7.2.3.11 with Corollary 4.6.6.25, we deduce that the inclusion map

\[ \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \hookrightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \]

is right anodyne.

Proposition 7.2.3.13. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d] \ar [r]^-{F} & \operatorname{\mathcal{C}}\ar [d]^-{\pi } \\ \operatorname{\mathcal{D}}' \ar [r]^-{\overline{F}} & \operatorname{\mathcal{D}}. } \]

If $\pi $ is a cocartesian fibration and $\overline{F}$ is right cofinal, then $F$ is right cofinal.

Proof. By virtue of Corollary 7.2.1.14, 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

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \times _{\Delta ^ n} \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {G}) \ar [r] \ar [d]^{F'} & \Lambda ^{n}_{i} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\ar [d]^{F} \\ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {G}) \ar [r]^-{\lambda } & \operatorname{\mathcal{C}}, } \]

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 7.2.1.21). $\square$

Corollary 7.2.3.14. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d] \ar [r]^-{F} & \operatorname{\mathcal{C}}\ar [d]^-{\pi } \\ \operatorname{\mathcal{D}}' \ar [r]^-{\overline{F}} & \operatorname{\mathcal{D}}. } \]

If $\pi $ is a cocartesian fibration and $\overline{F}$ is right anodyne, then $F$ is right anodyne.

Example 7.2.3.15. Let $\pi : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories, let $X$ be an object of $\operatorname{\mathcal{D}}$, and set $\operatorname{\mathcal{C}}_{X} = \{ X\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. If $X$ is a final object of $\operatorname{\mathcal{D}}$, then the inclusion map $\operatorname{\mathcal{C}}_{X} \hookrightarrow \operatorname{\mathcal{C}}$ is right anodyne, and therefore right cofinal. This follows by combining Corollaries 7.2.3.14 and 4.6.6.25.

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 4.6.6.23), 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

\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{C}}' \xrightarrow {F''} \operatorname{\mathcal{D}}, \]

where $F'$ is inner anodyne and $F''$ is an inner fibration. Since $F'$ is left 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

\[ \operatorname{\mathcal{D}}\xleftarrow { \widetilde{\operatorname{ev}}_0 } \operatorname{\mathcal{M}}\xrightarrow { \widetilde{ \operatorname{ev}}_{1} } \operatorname{\mathcal{C}}, \]

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

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times \operatorname{\mathcal{C}}\ar [r] \ar [d]^{\widetilde{\delta }} & (\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ (\{ 1\} \times \operatorname{\mathcal{C}}) } (\{ 1\} \times \operatorname{\mathcal{M}}) \ar [r] \ar [d]^{\iota } & \operatorname{\mathcal{C}}\ar [d]^{ \widetilde{\delta } } \\ \{ 0\} \times \operatorname{\mathcal{M}}\ar [r] & \Delta ^1 \times \operatorname{\mathcal{M}}\ar [r]^-{ \widetilde{h} } & \operatorname{\mathcal{M}}, } \]

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 (Proposition 5.3.7.1). 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 5.1.6.7), 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 7.2.1.9). We now observe that the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ factors as a composition

\[ \operatorname{\mathcal{C}}\xrightarrow { \widetilde{\delta } } \operatorname{\mathcal{M}}\xrightarrow { \widetilde{\operatorname{ev}}_1 } \operatorname{\mathcal{D}}, \]

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

Combining Theorem 7.2.3.1 with Proposition 7.2.3.13, we obtain the following:

Corollary 7.2.3.16 (Fiberwise Cofinality Criterion). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{E}}' \ar [dr]^{U'} \ar [rr]^{ F } & & \operatorname{\mathcal{E}}\ar [dl]_{U} \\ & \operatorname{\mathcal{C}}& } \]

where $U$ and $U'$ are cartesian fibrations, and the morphism $F$ carries $U'$-cartesian edges of $\operatorname{\mathcal{E}}'$ to $U$-cartesian edges of $\operatorname{\mathcal{E}}$. The following conditions are equivalent:

$(1)$

The morphism $F$ is right cofinal.

$(2)$

For every vertex $C \in \operatorname{\mathcal{C}}$, the induced map of fibers $F_{C}: \operatorname{\mathcal{E}}'_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$ is right cofinal.

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 7.2.1.21). 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

\[ \operatorname{\mathcal{E}}'_{C} \times _{ \operatorname{\mathcal{E}}_{C} } (\operatorname{\mathcal{E}}_{C})_{X/} \hookrightarrow \operatorname{\mathcal{E}}' \times _{ \operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_{ X/ } \]

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

\[ \xymatrix { \operatorname{\mathcal{E}}'_{C} \times _{ \operatorname{\mathcal{E}}_{C} } (\operatorname{\mathcal{E}}_{C})_{X/} \ar [r] \ar [d] & \operatorname{\mathcal{E}}' \times _{ \operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_{ X/ } \ar [d] \\ \{ \operatorname{id}_{C} \} \ar [r] & \operatorname{\mathcal{C}}_{C/}, } \]

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 4.6.6.24). This is a special case of Proposition 4.6.6.23. $\square$