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5.3.7 Application: Path Fibrations

Recall that every morphism of Kan complexes $f: X \rightarrow Y$ admits a canonical factorization

$X \xrightarrow {\delta } P(f) \xrightarrow {\pi } Y,$

where $\delta$ is a homotopy equivalence and $\pi$ is the path fibration

$P(f) = X \times _{ \operatorname{Fun}( \{ 0\} , Y)} \operatorname{Fun}( \Delta ^1, Y) \rightarrow \operatorname{Fun}( \{ 1\} , Y) \simeq Y$

of Example 3.1.7.9. Note that the simplicial set $P(f) = X \operatorname{\vec{\times }}_{Y} Y$ is an example of an oriented fiber product (Definition 4.6.4.1), which is defined for any morphism of simplicial sets $f: X \rightarrow Y$. Beware that if $X$ and $Y$ are not Kan complexes, then $\delta$ need not be a homotopy equivalence and $\pi$ need not be a Kan fibration. However, if $X = \operatorname{\mathcal{C}}$ and $Y = \operatorname{\mathcal{D}}$ are $\infty$-categories, then we have the following weaker statements:

$(a)$

The functor $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is fully faithful, and its essential image is the homotopy fiber product $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ of Construction 4.5.2.1 (Proposition 5.3.7.4).

$(b)$

The functor $\pi : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is a cocartesian fibration of $\infty$-categories (Proposition 5.3.7.1).

Moreover, the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ can be characterized by a universal mapping property: roughly speaking, the diagonal map $\delta$ exhibits the cocartesian fibration $\pi$ as freely generated by the functor $f$ (Theorem 5.3.7.7).

Proposition 5.3.7.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets and let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the oriented fiber product of Definition 4.6.4.1, so that evaluation at the vertices $0,1 \in \Delta ^1$ determines maps

$\operatorname{\mathcal{C}}\xleftarrow {\pi '} \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\xrightarrow {\pi } \operatorname{\mathcal{D}}.$

Then:

$(1)$

If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ are $\infty$-categories, then the evaluation map $\pi : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is a cocartesian fibration of simplicial sets. Moreover, an edge $e$ of the simplicial set $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is $\pi$-cocartesian if and only if $\pi '(e)$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$.

$(2)$

If $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are $\infty$-categories, then the evaluation map $\pi ': \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibration of simplicial sets. Moreover, an edge $e$ of the simplicial set $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is $\pi '$-cartesian if and only if $\pi (e)$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$.

Example 5.3.7.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Applying Proposition 5.3.7.1 in the case where both $F$ and $G$ are the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$, we deduce that the evaluation functor

$\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}$

is a cartesian fibration of $\infty$-categories, and the evaluation functor

$\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}$

is a cocartesian fibration of $\infty$-categories.

Proof of Proposition 5.3.7.1. We will prove assertion $(1)$; the proof of $(2)$ is similar. Note that the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ fits into a pullback diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [d]^-{\pi } \ar [r] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\ar [d] \\ \operatorname{\mathcal{D}}\ar [r]^-{G} & \operatorname{\mathcal{E}}. }$

By virtue of Remark 5.1.4.6, we can replace $\operatorname{\mathcal{D}}$ by $\operatorname{\mathcal{E}}$ and thereby reduce to the case where $\operatorname{\mathcal{D}}= \operatorname{\mathcal{E}}$ and $G$ is the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$. It follows from Proposition 4.6.4.2 that the map $(\pi ', \pi ): \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration of $\infty$-categories, so that $\pi : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is also an isofibration. Let us say that a morphism $e$ of the $\infty$-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special if $\pi '(e)$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$. Proposition 5.3.7.1 is an immediate consequence of the following three assertions:

$(a)$

For every object $x$ in the $\infty$-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ and every morphism $\overline{e}: \pi (x) \rightarrow \overline{y}$ in the $\infty$-category $\operatorname{\mathcal{D}}$, there exists a special morphism $e: x \rightarrow y$ in $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying $\pi (e) = \overline{e}$.

$(b)$

Every special morphism of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is $\pi$-cocartesian.

$(c)$

Every $\pi$-cocartesian morphism of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special.

We begin with the proof of $(a)$. Let $x$ be an object of the $\infty$-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, which we identify with a triple $(\overline{x}', \overline{x}, u)$ where $\overline{x}' = \pi '(x)$ is an object of $\operatorname{\mathcal{C}}$, $\overline{x} = \pi (x)$ is an object of $\operatorname{\mathcal{D}}$, and $u: F( \overline{x}' ) \rightarrow \overline{x}$ is a morphism of $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ is an $\infty$-category, we can choose a $2$-simplex $\sigma$ of $\operatorname{\mathcal{E}}$ satisfying $d_0(\sigma ) = \overline{e}$ and $d_2(\sigma ) = f$. Set $g = d_1(\sigma )$. Then the $2$-simplices $\sigma$ and $s_0(g)$ together determine a commutative diagram

$\xymatrix@R =50pt@C=50pt{ F( \overline{x}' ) \ar [d]_-{ \operatorname{id}_{ F( \overline{x}' ) }} \ar [r]^-{f} \ar [dr]_{g} & \overline{x} \ar [d]^-{ \overline{e} } \\ F( \overline{x}' ) \ar [r]^-{g} & \overline{y} }$

in the $\infty$-category $\operatorname{\mathcal{D}}$, which we can identify with an edge $e: x \rightarrow y$ of the $\infty$-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying $\pi '(e) = \operatorname{id}_{ \overline{x}'}$ and $\pi (e) = \overline{e}$.

We now prove $(b)$. Let $n \geq 2$ and suppose that we are given a lifting problem

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{ \tau _0 } \ar [r] \ar [d] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [d]^-{ \pi } \\ \Delta ^{n} \ar [r]^-{ \overline{\tau } } \ar@ {-->}[ur]^{\tau } & \operatorname{\mathcal{D}}}$

for which the composite map

$\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \hookrightarrow \Lambda ^{n}_0 \xrightarrow { \tau _0 } \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$

is a special edge of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. Then $\pi ' \circ \tau _0$ carries the initial edge of $\Lambda ^{n}_0$ to an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$, and can therefore be extended to an $n$-simplex $\rho : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$. Let $X(0)$ denote the simplicial subset of $\Delta ^1 \times \Delta ^ n$ given by the union of $\{ 0 \} \times \Delta ^ n$, $\{ 1\} \times \Delta ^ n$, and $\Delta ^1 \times \Lambda ^{n}_0$. The morphisms $\rho$, $\overline{\tau }$, and $\tau _0$ can then be amalgamated to a morphism of simplicial sets $h_0: X(0) \rightarrow \operatorname{\mathcal{D}}$. We wish to show that $h_0$ can be extended to a map $h: \Delta ^1 \times \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$. Choose a filtration

$X(0) \subset X(1) \subset X(2) \subset \cdots \subset X(t) = \Delta ^{1} \times \Delta ^{n}$

satisfying the requirements of Lemma 4.4.4.7. We will complete the proof of $(b)$ by showing that, for each $s \leq t$, the morphism $h_0$ admits an extension $h_ s: X(s) \rightarrow \operatorname{\mathcal{D}}$. The proof proceeds by induction on $s$, the case $s = 0$ being vacuous. Let us therefore assume that $0 < s \leq t$ and that $h_0$ has already been extended to a morphism of simplicial sets $h_{s-1}: X(s-1) \rightarrow \operatorname{\mathcal{D}}$. By construction, we have a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{q}_{p} \ar [r]^-{\varphi } \ar [d] & X(s-1) \ar [d] \\ \Delta ^{q} \ar [r] & X(s) }$

for some $q \geq 2$ and $0 \leq p < q$. Consequently, to prove the existence of $h_ s$, it will suffice to show that $h_{s-1} \circ \varphi$ can be extended to a $q$-simplex of $\operatorname{\mathcal{D}}$. For $p \neq 0$, the existence of this extension follows from our assumption that $\operatorname{\mathcal{D}}$ is an $\infty$-category. In the case $p = 0$, Lemma 4.4.4.7 guarantees that the morphism $\varphi$ carries the initial edge of $\Delta ^{q}$ to the edge $(0,0) \rightarrow (0,1)$ of $\Delta ^{1} \times \Delta ^ n$, so that $h_{s-1} \circ \varphi$ carries the initial edge of $\Delta ^ q$ to an isomorphism in $\operatorname{\mathcal{D}}$. In this case, the existence of the desired extension follows from Theorem 4.4.2.6.

We now prove $(c)$. Let $e: x \rightarrow z$ be a $\pi$-cocartesian morphism in the $\infty$-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$; we wish to show that $e$ is special. By virtue of $(a)$, there exists a special morphism $e': x \rightarrow y$ of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying $\pi (e) = \pi (e')$. It follows from $(b)$ that $e'$ is also $\pi$-cocartesian. Applying Remark 5.1.3.8, we deduce that there exists a commutative diagram

$\xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{ e'' } & \\ x \ar [ur]^{e'} \ar [rr]^-{e} & & z }$

in the $\infty$-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, where $e''$ is an isomorphism. Then $\pi '(e')$ and $\pi '(e'')$ are isomorphisms in the $\infty$-category $\operatorname{\mathcal{C}}$, so that $\pi '(e)$ is also an isomorphism in $\operatorname{\mathcal{C}}$ (Remark 1.3.6.3). It follows that $e$ is a special morphism of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, as desired. $\square$

Corollary 5.3.7.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $K$ be a simplicial set. Then:

$(1)$

The restriction map $U: \operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is a cocartesian fibration. Moreover, a morphism $e$ of $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}})$ is $U$-cocartesian if and only if it carries the cone point ${\bf 0} \in K^{\triangleleft }$ to an isomorphism in $\operatorname{\mathcal{C}}$.

$(2)$

The restriction map $V: \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is a cartesian fibration. Moreover, a morphism $e$ of $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ is $U$-cartesian if and only if it carries the cone point ${\bf 1} \in K^{\triangleright }$ to an isomorphism in $\operatorname{\mathcal{C}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $\Delta ^0 \diamond K$ denote the blunt join of Notation 4.5.8.3, and let $c: \Delta ^0 \diamond K \rightarrow \Delta ^{0} \star K = K^{\triangleleft }$ be the categorical equivalence of Theorem 4.5.8.8. We have a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}}) \ar [dr]^{U} \ar [rr]^-{ \circ c} & & \operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}}) \ar [dl]^{U'} \\ & \operatorname{Fun}(K, \operatorname{\mathcal{C}}) & }$

where the horizontal map is an equivalence of $\infty$-categories (Proposition 4.5.3.8) and the vertical maps are isofibrations (Corollary 4.4.5.3). Unwinding the definitions, we can identify $\operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}})$ with the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Under this identification, the functor $U'$ is given by projection onto the second factor, and is therefore a cocartesian fibration (Proposition 5.3.7.1). Applying Corollary 5.1.5.2, we deduce that $U$ is also a cocartesian fibration. Moreover, a morphism $e$ of $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}})$ is $U$-cocartesian if and only if its image in $\operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}})$ is $U'$-cocartesian (Proposition 5.1.5.6). Using the criterion of Proposition 5.3.7.1, we see that this is equivalent to the requirement that $e$ carries the cone point ${\bf 0} \in K^{\triangleleft }$ to an isomorphism in $\operatorname{\mathcal{C}}$. $\square$

Proposition 5.3.7.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let

$\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\{ 0\} , \operatorname{\mathcal{D}})} \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$

be map induced by the diagonal embedding $c: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$. Then $\delta$ is fully faithful, and its essential image is the homotopy fiber product $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ of Construction 4.5.2.1.

Proof. Let us identify the objects of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ with triples $(C,D,u)$, where $C$ is an object of $\operatorname{\mathcal{C}}$, $D$ is an object of $\operatorname{\mathcal{D}}$, and $u: F(C) \rightarrow D$ is a morphism in $\operatorname{\mathcal{D}}$. By definition, $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is the full subcategory of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ spanned by those triples $(C,D,u)$ where $u$ is an isomorphism in $\operatorname{\mathcal{D}}$. The functor $\delta$ is given on objects by the formula $\delta (C) = ( C, F(C), \operatorname{id}_{ F(C) } )$, and therefore factors through $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. To complete the proof, it will suffice to show that the functor $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is an equivalence of $\infty$-categories. Equivalently, we wish to show that the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^{\operatorname{id}} \ar [d]^{F} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}\ar [r]^{\operatorname{id}} & \operatorname{\mathcal{D}}}$

is a categorical pullback square, which is a special case of Proposition 4.5.2.19. $\square$

Corollary 5.3.7.5. Let $f: K \rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty$-category. Then $f$ factors as a composition $K \xrightarrow {j} \operatorname{\mathcal{C}}\xrightarrow {U} \operatorname{\mathcal{D}}$, where $U$ is an isofibration of $\infty$-categories and $j$ is both a monomorphism and a categorical equivalence.

Proof. Using Proposition 4.1.3.2, we can factor $f$ as a composition $K \xrightarrow {i} \operatorname{\mathcal{K}}\xrightarrow {F} \operatorname{\mathcal{D}}$, where $i$ is inner anodyne and $F$ is an inner fibration. Note that the simplicial set $\operatorname{\mathcal{K}}$ is an $\infty$-category (Remark 4.1.1.9), and that $i$ is a categorical equivalence of simplicial sets (Corollary 4.5.3.14). We may therefore replace $f$ by $F$, and thereby reduce to the special case where $K = \operatorname{\mathcal{K}}$ is an $\infty$-category. Let $\operatorname{\mathcal{C}}$ denote the homotopy fiber product $\operatorname{\mathcal{K}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. Then $F$ factors as a composition

$\operatorname{\mathcal{K}}\xrightarrow { \delta } \operatorname{\mathcal{K}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\xrightarrow {U} \operatorname{\mathcal{D}},$

where the diagonal embedding $\delta$ is an equivalence of $\infty$-categories (Proposition 5.3.7.4) and $U$ is an isofibration (see Remark 4.5.2.2). $\square$

Remark 5.3.7.6. In the situation of Corollary 5.3.7.5, it is not necessary to assume that $\operatorname{\mathcal{D}}$ is an $\infty$-category: every morphism of simplicial sets $f: X \rightarrow Z$ admits a factorization $X \xrightarrow {f'} Y \xrightarrow {f''} Z$, where $f''$ is an isofibration and $f'$ both a monomorphism and a categorical equivalence (Proposition ). However, the proof is somewhat more difficult.

Theorem 5.3.7.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $\pi : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ be given by projection onto the second factor, let $\delta : \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ be the diagonal map. For every cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$, precomposition with $\delta$ induces a trivial Kan fibration of $\infty$-categories

$\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}).$

To prove Theorem 5.3.7.7, we will need a variant of the direct image construction studied in §4.5.9.

Notation 5.3.7.8 (Cocartesian Direct Images). Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be morphisms of simplicial sets, and let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})$ be the simplicial set of Construction 4.5.9.1. Using Remark 4.5.9.8, we can identify vertices of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})$ with pairs $(C, F)$, where $C$ is a vertex of $\operatorname{\mathcal{C}}$ and

$F: \operatorname{\mathcal{D}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}= \operatorname{\mathcal{E}}_{C}$

is a section of the map $V|_{\operatorname{\mathcal{E}}_ C}: \operatorname{\mathcal{E}}_ C \rightarrow \operatorname{\mathcal{D}}_{C}$. If $V$ is a cocartesian fibration, we let $\operatorname{Res}^{\operatorname{CCart}}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})$ denote the full simplicial subset of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ spanned by those vertices $(C,F)$ where $F$ carries each edge of $\operatorname{\mathcal{D}}_{C}$ to $V_{C}$-cocartesian edge of $\operatorname{\mathcal{E}}_{C}$. We will refer to $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}})$ as the cocartesian direct image of $\operatorname{\mathcal{E}}$ along $U$.

Remark 5.3.7.9. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets and let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of simplicial sets. Then the projection map $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ restricts to a projection map $\pi ^{\operatorname{CCart}}: \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$. Moreover, for each vertex $C \in \operatorname{\mathcal{C}}$, the isomorphism $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_ C }( \operatorname{\mathcal{D}}_ C, \operatorname{\mathcal{E}}_ C )$ of Remark 4.5.9.8 restricts to an isomorphism of full subcategories $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}) \simeq \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{D}}_ C }( \operatorname{\mathcal{D}}_ C, \operatorname{\mathcal{E}}_ C )$.

Proposition 5.3.7.10. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of simplicial sets, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets. Then:

$(1)$

The projection map $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of simplicial sets.

$(2)$

Let $e: X \rightarrow Y$ be a $\pi$-cocartesian edge of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$. If $X$ belongs to the simplicial subset $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}})$, then $Y$ also belongs to the simplicial subset $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}})$.

$(3)$

The morphism $\pi$ restricts to a cocartesian fibration $\pi ^{\operatorname{CCart}}: \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$.

$(4)$

An edge of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}})$ is $\pi ^{\operatorname{CCart}}$-cocartesian if and only if it is $\pi$-cocartesian.

Proof. Assertion $(1)$ follows from Proposition 5.3.6.6 (after passing to opposite simplicial sets). To prove $(2)$, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$ and $\pi (e)$ is the nondegenerate edge of $\operatorname{\mathcal{C}}$. In this case, the simplicial sets $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are $\infty$-categories, and we can identify the edge $e$ with a morphism of simplicial sets $E: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ satisfying $V \circ E = \operatorname{id}_{\operatorname{\mathcal{D}}}$. Let $u: D \rightarrow D'$ be a morphism in the $\infty$-category $\operatorname{\mathcal{D}}_1 = \{ 1\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$; we wish to show that $E(u)$ is a $V$-cocartesian morphism of $\operatorname{\mathcal{E}}$. To prove this, let $G: \operatorname{\mathcal{D}}_{1} \rightarrow \operatorname{\mathcal{D}}_0 = \{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ be given by contravariant transport along the nondegenerate edge of $\operatorname{\mathcal{C}}$, so that we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ G(D) \ar [d]^{ G(u) } \ar [r] & D \ar [d]^{u} \\ G(D') \ar [r] & D', }$

in the $\infty$-category where the horizontal maps are $U$-cartesian. Our assumption that $e$ is $\pi$-cocartesian guarantees that the functor $E$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{D}}$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ (Proposition 5.3.6.6). We therefore obtain a commutative diagram

$\xymatrix@R =50pt@C=50pt{ (E \circ G)(D) \ar [d]^{ (E \circ G)(u) } \ar [r] & E(D) \ar [d]^{ E(u) } \\ (E \circ G)(D') \ar [r] & E(D' ), }$

where the horizontal maps are $V$-cocartesian. By virtue of Corollary 5.1.2.4, it will suffice to show that the morphism $(E \circ G)(u)$ is $V$-cocartesian, which follows from our assumption that $X$ belongs to $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}})$. This completes the proof of $(2)$; assertions $(3)$ and $(4)$ then follow by applying Proposition 5.1.4.16. $\square$

In the situation of Proposition 5.3.7.10, the cocartesian direct image $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}})$ can be characterized by a universal property:

Proposition 5.3.7.11. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of simplicial sets and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets. For every cocartesian fibration of simplicial sets $W: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, the isomorphism

$\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) ) \xrightarrow {\sim } \operatorname{Fun}_{/\operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$

of Remark 4.5.9.3 restricts to an isomorphism of full simplicial subsets

$\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}) ) \xrightarrow {\sim } \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}).$

Proof. Let $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ denote the projection map and let $f: \operatorname{\mathcal{C}}' \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ be a morphism satisfying $\pi \circ f = W$, corresponding to a morphism of simplicial sets $F: \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ for which $V \circ F$ is given by projection to the second factor. Note that we can regard $F$ as a vertex of the simplicial subset $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ if and only if it satisfies the following condition:

$(a)$

For every edge $(e',e)$ of the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ for which $e'$ is a $W$-cocartesian edge of $\operatorname{\mathcal{C}}'$, the image $F(e',e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$.

We wish to show that $(a)$ is equivalent to the following pair of conditions:

$(b)$

The morphism $f$ factors through the full simplicial subset $\operatorname{Res}^{\operatorname{CCart}}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \subseteq \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$. In other words, for every edge $(e',e)$ of the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ for which $e'$ is a degenerate edge of $\operatorname{\mathcal{C}}'$, the image $F(e',e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$.

$(c)$

For every $W$-cocartesian edge $e'$ of $\operatorname{\mathcal{C}}'$, the image $f(e')$ is a $\pi |_{ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}) }$-cocartesian edge of $\operatorname{Res}^{\operatorname{CCart}}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$. By virtue of Propositions 5.3.7.10 and 5.3.6.6, this is equivalent to the assertion that for every edge $(e',e)$ of the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ where $e'$ is $W$-cocartesian and $e$ is $U$-cartesian, the image $F(e',e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$.

The implications $(a) \Rightarrow (b)$ and $(a) \Rightarrow (c)$ are clear. For the converse, suppose that $(b)$ and $(c)$ are satisfied; we wish to prove $(a)$. Let $(e',e): (X',X) \rightarrow (Z',Z)$ be an edge of the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, where $e': X' \rightarrow Z'$ is $W$-cocartesian. Let $\overline{e} = U(e) = W(e')$ denote the corresponding edge of $\operatorname{\mathcal{C}}$. Since $U$ is a cartesian fibration, there exists a $U$-cartesian morphism $f: Y \rightarrow Z$ satisfying $U(f) = \overline{e}$. Let $\overline{\sigma }$ denote the left-degenerate $2$-simplex $s_0(\overline{e})$. Since $f$ is $U$-cartesian, we can lift $\overline{\sigma }$ to a $2$-simplex of $\operatorname{\mathcal{D}}$ as indicated in the diagram

$\xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{f} & \\ X \ar [ur] \ar [rr]^-{e} & & Z. }$

Writing $\sigma '$ for the left-degenerate $2$-simplex $s_0(e')$ of $\operatorname{\mathcal{C}}'$, we obtain a $2$-simplex $\tau = F( \sigma ', \sigma )$ of $\operatorname{\mathcal{E}}$. It follows from assumption $(b)$ that the restriction $\tau |_{ \operatorname{N}_{\bullet }( \{ 0 < 1\} ) }$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$, and from assumption $(c)$ that the restriction $\tau |_{ \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) }$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$. Applying Proposition 5.1.4.12, we conclude that $F(e',e) = \tau |_{ \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) }$ is also a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$. $\square$

Proof of Theorem 5.3.7.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty$-categories, and let $\delta : \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ be the diagonal embedding. Since $U$ is an isofibration (Proposition 5.1.4.8), the restriction map $\overline{\theta }: \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is also an isofibration (Corollary 4.5.5.16). Because $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ is a replete full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$, it follows that $\overline{\theta }$ restricts to an isofibration $\theta : \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. To prove Theorem 5.3.7.7, we will show that $\theta$ is an equivalence of $\infty$-categories (it is then automatically a trivial Kan fibration of simplicial sets: see Proposition 4.5.5.20).

Note that the functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ induces cocartesian fibrations $U': \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ and $U'': \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Let $\pi ': \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be given by projection onto the first factor, so that $\pi '$ is a cartesian fibration (Proposition 5.3.7.1). Let $\operatorname{\mathcal{M}}$ denote the cocartesian direct image $\operatorname{Res}^{\operatorname{CCart}}_{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}})$ and let $T: \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{C}}$ be the projection map. Precomposition with the diagonal embedding $\delta : \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ induces a restriction functor

$\delta ^{\ast }: \operatorname{\mathcal{M}}\rightarrow \operatorname{Res}_{ \operatorname{\mathcal{C}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}) = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$

which fits into a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{M}}\ar [rr]^{ \delta ^{\ast } } \ar [dr]_{T} & & \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}\ar [dl]^{U''} \\ & \operatorname{\mathcal{C}}& }$

It follows from Proposition 5.3.7.10 that $T$ is a cocartesian fibration and that $\delta ^{\ast }$ carries $T$-cocartesian morphisms of $\operatorname{\mathcal{M}}$ to $U''$-cocartesian morphisms of $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$. Using Proposition 5.3.7.11, we can identify $\theta$ with the map

$\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{M}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}, \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$

given by postcomposition with $\delta ^{\ast }$. Consequently, to show that $\theta$ is an equivalence of $\infty$-categories, it will suffice to show that $\delta ^{\ast }$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$. By virtue of Proposition 5.1.6.14), this can be checked fiberwise: that is, it suffices to show that for each object $C \in \operatorname{\mathcal{C}}$, the induced map of fibers

$\delta ^{\ast }_{C}: \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{M}}\simeq \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{D}}}( \{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \{ C\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$

is an equivalence of $\infty$-categories. This is a special case of Corollary 5.3.1.22, since $\delta (C)$ is an initial object of the $\infty$-category $\{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ (Proposition 4.6.6.23). $\square$