# Kerodon

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### 4.6.5 Oriented Fiber Products

Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be categories. To every pair of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, one can associate the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, whose objects are triples $(C, D, \eta )$ where $C$ is an object of $\operatorname{\mathcal{C}}$, $D$ is an object of $\operatorname{\mathcal{D}}$, and $\eta : F(C) \rightarrow G(D)$ is a morphism in the category $\operatorname{\mathcal{E}}$ (Notation 2.1.4.19). This construction has an obvious analogue in the setting of $\infty$-categories. We now generalize this construction to the setting of $\infty$-categories.

Definition 4.6.5.1 (The Oriented Fiber Product). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. We let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the simplicial set given by the iterated fiber product

$\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{E}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}(\{ 1\} , \operatorname{\mathcal{E}}) } \operatorname{\mathcal{D}}.$

We will refer to $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ as the oriented fiber product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ over $\operatorname{\mathcal{E}}$.

As our notation suggests, we will be primarily interested in the special case of Definition 4.6.5.1 where the simplicial sets $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are $\infty$-categories.

Proposition 4.6.5.2. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category, and suppose we are given morphisms of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Then the projection map $\theta : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration of simplicial sets.

Proof. By construction, we have a pullback diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [r] \ar [d]^{\theta } & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \ar [d]^{\theta _0} \\ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{E}}). }$

Since $\operatorname{\mathcal{E}}$ is an $\infty$-category, the restriction map $\theta _0$ is an isofibration of $\infty$-categories (Corollary 4.4.5.3). Invoking Remark 4.5.7.7, we conclude that $\theta$ is an isofibration of simplicial sets. $\square$

Corollary 4.6.5.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories. Then the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty$-category.

Proof. By virtue of Proposition 4.6.5.2, the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration. Since $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an $\infty$-category, it follows that $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty$-category (Remark 4.5.7.3). $\square$

Example 4.6.5.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors between ordinary categories, and let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})$ denote the oriented fiber product of Notation 2.1.4.19. Since the nerve construction is compatible with the formation of inverse limits and functor categories, we have a canonical isomorphism of simplicial sets

$\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}) \simeq ( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})} \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ).$

Consequently, Definition 4.6.5.1 can be viewed as a generalization of the classical comma construction in category theory.

Example 4.6.5.5. Let $X$, $Y$, and $Z$ be topological spaces, and suppose we are given a pair of continuous functions $f: X \rightarrow Z$ and $g: Y \rightarrow Z$. We let $X \times ^{h}_{Z} Y$ denote the set of all triples $(x,y,\eta )$ where $x$ is a point of $X$, $y$ is a point of $Y$, and $\eta : [0,1] \rightarrow Z$ is a continuous function satisfying $\eta (0) = f(x)$ and $\eta (1) = g(y)$. We will refer to $X \times ^{h}_{Z} Y$ as the homotopy fiber product of $X$ and $Y$ over $Z$. The homotopy fiber product $X \times ^{h}_{Z} Y$ carries a natural topology, which is defined by viewing it as a subspace of the product $X \times Y \times \operatorname{Hom}_{\operatorname{Top}}( [0,1], Z)$ (where we endow the path space $\operatorname{Hom}_{\operatorname{Top}}([0,1],Z)$ with the compact-open topology). We then have a canonical isomorphism of simplicial sets

$\operatorname{Sing}_{\bullet }( X \times _{Z}^{h} Y) \simeq \operatorname{Sing}_{\bullet }(X) \operatorname{\vec{\times }}_{ \operatorname{Sing}_{\bullet }(Z) } \operatorname{Sing}_{\bullet }(Y)$

where the right hand side is the oriented fiber product of Definition 4.6.5.1.

Example 4.6.5.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets. Then the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\Delta ^0} \operatorname{\mathcal{D}}$ can be identified with the cartesian product $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$.

Remark 4.6.5.7. 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 $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$ and $G^{\operatorname{op}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$ be the opposite morphisms. Then we have a canonical isomorphism of simplicial sets

$(\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})^{\operatorname{op}} \simeq (\operatorname{\mathcal{D}}^{\operatorname{op}} \operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}^{\operatorname{op}}} \operatorname{\mathcal{C}}^{\operatorname{op}} ).$

Remark 4.6.5.8. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $X$ be a vertex of $\operatorname{\mathcal{C}}$. For any simplicial set $K$, we have canonical isomorphisms of simplicial sets

$\operatorname{Fun}(K, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} ) \simeq \operatorname{Fun}_{ \{ x\} / }( K \diamond \{ x\} , \operatorname{\mathcal{C}}) \quad \quad \operatorname{Fun}(K, \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}_{ \{ x\} / }( \{ x\} \diamond K, \operatorname{\mathcal{C}}),$

where $K \diamond \{ x\}$ and $\{ x\} \diamond K$ denote the simplicial sets defined in Notation 4.5.5.3. Restricting to vertices, we obtain bijections

$\{ \textnormal{Morphisms K \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} } \} \simeq \{ \textnormal{Morphisms f: (K \diamond \{ x\} ) \rightarrow \operatorname{\mathcal{C}} with f(x) = X} \}$
$\{ \textnormal{Morphisms K \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}} \} \simeq \{ \textnormal{Morphisms f: (\{ x\} \diamond K) \rightarrow \operatorname{\mathcal{C}} with f(x) = X} \} .$

Example 4.6.5.9. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, which we identify with morphisms of simplicial sets $X,Y: \Delta ^{0} \rightarrow \operatorname{\mathcal{C}}$. Then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 4.6.1.1 is the oriented fiber product $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\}$.

The following result is a relative version of Proposition 4.6.1.8:

Proposition 4.6.5.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing an object $X$. Then the projection map $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration and the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.

Proof. We will prove the second assertion; the first follows by a similar argument. Let $A \hookrightarrow B$ be a right anodyne morphism of simplicial sets; we wish to show that every lifting problem

$\xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}}$

admits a solution. Unwinding the definitions, we are reduced to showing that a map of simplicial sets

$\sigma _0: B \coprod _{A} (A \diamond \{ X\} ) \rightarrow \operatorname{\mathcal{C}}$

can be extended to a map $\sigma : B \diamond \{ X\} \rightarrow \operatorname{\mathcal{C}}$ (see Notation 4.5.5.3). By virtue of Lemma 4.5.6.2, it will suffice to show that the inclusion map

$\iota : B \coprod _{A} ( A \diamond \{ X\} ) \hookrightarrow B \diamond \{ X\}$

is a categorical equivalence of simplicial sets, which follows from Corollary 4.5.5.13. $\square$

Corollary 4.6.5.11. Let $\operatorname{\mathcal{D}}$ be an $\infty$-category containing an object $X$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then the projection map $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration, and the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.

Proof. Unwinding the definition, we have pullback diagrams

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

The desired result now follows by combining Proposition 4.6.5.10 with Remark 4.2.1.8. $\square$

Recall that, if $\operatorname{\mathcal{C}}$ is an ordinary category containing an object $X$, then the slice and coslice categories $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}_{X/}$ can be identified with the oriented fiber products $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\}$ and $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}})$, respectively (Remark 4.3.1.7). Our next goal is to establish a similar result in the setting of $\infty$-categories. Here the situation is a bit more subtle: if $X$ is an object of an $\infty$-category $\operatorname{\mathcal{C}}$, then the simplicial sets $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\}$ are generally not isomorphic. However, we will show that they are nevertheless equivalent as $\infty$-categories.

Construction 4.6.5.12. For each $n \geq 0$, let

$c^{\mathrm{L}}_{n}: \Delta ^1 \times \Delta ^{n} \simeq \Delta ^{0} \times \Delta ^1 \times \Delta ^{n} \rightarrow \Delta ^{0} \star \Delta ^{n} \simeq \Delta ^{n+1}$
$c^{\mathrm{R}}_{n}: \Delta ^{n} \times \Delta ^{1} \simeq \Delta ^{n} \times \Delta ^1 \times \Delta ^{0} \rightarrow \Delta ^{n} \star \Delta ^{0} \simeq \Delta ^{n+1}$

denote the collapse morphisms given by Construction 4.5.5.1, given concretely on vertices by the formulae

$c^{\mathrm{L}}_{n}(i,j) = \begin{cases} 0 & \textnormal{ if } i=0 \\ j+1 & \textnormal{ if } i=1 \end{cases} \quad \quad c^{\mathrm{R}}_{n}(j,i) = \begin{cases} j & \textnormal{ if } i = 0 \\ n+1 & \textnormal{ if } i = 1.\end{cases}$

Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a vertex $X$, and let $\sigma$ be an $n$-simplex of the coslice simplicial set $\operatorname{\mathcal{C}}_{X/}$, which we identify with a morphism of simplicial sets $\overline{\sigma }: \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{\sigma }(0)=X$. The composite map $\Delta ^{1} \times \Delta ^{n} \xrightarrow { c^{\mathrm{L}}_{n} } \Delta ^{n+1} \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}}$ can be identified with an $n$-simplex of the oriented fiber product $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, which we will denote by $\iota ^{\mathrm{L}}_{X}(\sigma )$. The construction $\sigma \mapsto \iota ^{\mathrm{L}}_{X}(\sigma )$ determines a morphism of simplicial sets $\iota ^{\mathrm{L}}_{X}: \operatorname{\mathcal{C}}_{X/} \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, which we will refer to as the left-pinch inclusion map.

Similarly, let $\tau$ be an $n$-simplex of the slice simplicial set $\operatorname{\mathcal{C}}_{/X}$, which we identify with a morphism of simplicial sets $\overline{\tau }: \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{\tau }(n+1) = X$. The composite map $\Delta ^ n \times \Delta ^1 \xrightarrow { c^{\mathrm{R}}_{n} } \Delta ^{n+1} \xrightarrow { \overline{\tau } } \operatorname{\mathcal{C}}$ can be identified with an $n$-simplex of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\}$ which we will denote by $\iota ^{\mathrm{R}}_{Y}(\tau )$. The construction $\sigma \mapsto \iota ^{\mathrm{R}}_{X}(\tau )$ determines a morphism of simplicial sets $\iota ^{\mathrm{R}}_{X}: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\}$, which we will refer to as the right-pinch inclusion map.

Remark 4.6.5.13. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a vertex $X$. Then the pinch inclusion maps

$\iota ^{\mathrm{L}}_{X}: \operatorname{\mathcal{C}}_{X/} \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\quad \quad \iota ^{\mathrm{R}}_{X}: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\}$

are monomorphisms of simplicial sets: this follows immediately from surjectivity of the collapse morphisms

$c_{n}^{\mathrm{L}}: \Delta ^1 \times \Delta ^{n} \twoheadrightarrow \Delta ^{n+1} \quad \quad c_{n}^{\mathrm{R}}: \Delta ^{n} \times \Delta ^{1} \twoheadrightarrow \Delta ^{n+1}.$

Theorem 4.6.5.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then the pinch inclusion maps

$\iota ^{\mathrm{L}}_{X}: \operatorname{\mathcal{C}}_{X/} \hookrightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\quad \quad \iota ^{\mathrm{R}}_{X}: \operatorname{\mathcal{C}}_{/X} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\}$

are equivalences of $\infty$-categories.

The proof of Theorem 4.6.5.14 will make use of the following:

Lemma 4.6.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $g: A \rightarrow \operatorname{\mathcal{C}}$ be a diagram indexed by a simplicial set $A$. Suppose we are given a pair of diagrams $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}_{g/}$ indexed by a simplicial set $B$, which we identify with diagrams $f'_0, f'_1: (A \star B) \rightarrow \operatorname{\mathcal{C}}$ satisfying $f'_0|_{A} = g = f'_{1}|_{A}$. The following conditions are equivalent:

$(1)$

The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the diagram $\infty$-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}}_{g/} )$.

$(2)$

The diagrams $f'_0$ and $f'_1$ are isomorphic when regarded as objects of the $\infty$-category $\operatorname{Fun}_{ A / }(A \star B, \operatorname{\mathcal{C}})$.

Proof. Choose a categorical mapping cylinder

$B \coprod B \xrightarrow {(s_0, s_1)} \overline{B} \xrightarrow {\pi } B$

for the simplicial set $B$ (Definition 4.6.4.3). Using Corollary 4.5.5.8, we deduce that the resulting diagram

$(A \star B) \coprod _{A} (A \star B) \xrightarrow {(s'_0, s'_1)} A \star \overline{B} \xrightarrow {\pi '} A \star B$

is a categorical mapping cylinder for the join $A \star B$ relative to $A$. Using the criterion of Corollary 4.6.4.11, we see that $(1)$ and $(2)$ can be reformulated as follows:

$(1')$

There exists a diagram $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}_{g/}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.

$(2')$

There exists a diagram $\overline{f}': A \star \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f'_0 = \overline{f}' \circ s'_0$ and $f'_1 = \overline{f}' \circ s'_1$.

The equivalence of $(1')$ and $(2')$ now follows immediately from the universal property of the coslice $\infty$-category $\operatorname{\mathcal{C}}_{g/}$. $\square$

Proof of Theorem 4.6.5.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing an object $X$. We will show that the left-pinch inclusion morphism $\iota ^{\mathrm{L}}_{X}: \operatorname{\mathcal{C}}_{X/} \hookrightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}})$ is an equivalence of $\infty$-categories; the analogous statement for $\iota ^{\mathrm{R}}_{X}$ follows by a similar argument. Let $K$ be a simplicial set and

$\rho : \pi _0( \operatorname{Fun}(K, \operatorname{\mathcal{C}}_{X/})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(K, \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}})^{\simeq } )$

be the map induced by composition with $\iota ^{\mathrm{L}}_{X}$; we wish to show that $\rho$ is a bijection. Using Lemma 4.6.5.15 and Remark 4.6.5.8, we can identify $\rho$ with the map

$\pi _0( \operatorname{Fun}_{\{ x\} / }( \{ x\} \star K, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}_{ \{ x\} }( \{ x\} \diamond K, \operatorname{\mathcal{C}})^{\simeq } )$

given by composition with the comparison map $c: \{ x\} \diamond K \rightarrow \{ x\} \star K$. We will complete the proof by showing that composition with $c$ induces an equivalence of $\infty$-categories $\operatorname{Fun}_{ \{ x\} / }( \{ x\} \star K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{ \{ x\} }( \{ x\} \diamond K, \operatorname{\mathcal{C}})$. This follows by applying Corollary 4.5.4.5 to the commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \{ x\} \star K, \operatorname{\mathcal{C}}) \ar [r]^-{\circ c} \ar [d] & \operatorname{Fun}( \{ x\} \diamond K, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( \{ x\} , \operatorname{\mathcal{C}}) \ar@ {=}[r] & \operatorname{Fun}( \{ x\} , \operatorname{\mathcal{C}}); }$

here the vertical maps are isofibrations (Corollary 4.4.5.3) and the upper horizontal map is an equivalence of $\infty$-categories by Proposition 4.5.2.7 (and the fact that $c$ is a categorical equivalence, by Theorem 4.5.5.7). $\square$

We conclude this section by describing another application of Lemma 4.6.5.15.

Proposition 4.6.5.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty$-categories and let $p: A \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the induced functors

$F_{p/}: \operatorname{\mathcal{C}}_{p/} \rightarrow \operatorname{\mathcal{D}}_{(F \circ p)/} \quad \quad F_{/p}: \operatorname{\mathcal{C}}_{/p} \rightarrow \operatorname{\mathcal{D}}_{/(F \circ p)}$

are also equivalences of $\infty$-categories.

Proof. We will prove that $F_{p/}$ is an equivalence of $\infty$-categories; the analogous assertion for $F_{/p}$ follows by a similar argument. Fix a simplicial set $B$; we wish to show that postcomposition with $F_{p/}$ induces a bijection

$\theta : \pi _0( \operatorname{Fun}(B, \operatorname{\mathcal{C}}_{p/} )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(A, \operatorname{\mathcal{D}}_{(F \circ p)/} )^{\simeq } ).$

By virtue of Lemma 4.6.5.15, we can identify $\theta$ with the map

$\pi _0( \operatorname{Fun}_{A/}( A \star B, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}_{A/}( A \star B, \operatorname{\mathcal{D}})^{\simeq } )$

induces by composition with $F$. It will therefore suffice to show that $F$ induces an equivalence of $\infty$-categories $\operatorname{Fun}_{A/}( A \star B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{A/}( A \star B, \operatorname{\mathcal{D}})$, which is a special case of Corollary 4.5.4.7. $\square$

Corollary 4.6.5.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be fully faithful functor of $\infty$-categories and let $p: A \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the induced functors

$F_{p/}: \operatorname{\mathcal{C}}_{p/} \rightarrow \operatorname{\mathcal{D}}_{(F \circ p)/} \quad \quad F_{/p}: \operatorname{\mathcal{C}}_{/p} \rightarrow \operatorname{\mathcal{D}}_{/(F \circ p)}$

are also fully faithful.

Proof. Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{D}}$ be the essential image of $F$ (Definition 4.6.2.9), so that $F$ induces an equivalence of $\infty$-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ (Corollary 4.6.2.19). By virtue of Proposition 4.6.5.16, the functor $F$ induces equivalences $\operatorname{\mathcal{C}}_{p/} \rightarrow \operatorname{\mathcal{C}}'_{ (F \circ p)/}$ and $F_{/p}: \operatorname{\mathcal{C}}_{/p} \rightarrow \operatorname{\mathcal{C}}'_{/(F \circ p)}$. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is a full subcategory of $\operatorname{\mathcal{D}}$ and $F$ is the inclusion map. In this case, the morphisms $F_{p/}$ and $F_{/p}$ are also the inclusions of full subcategories, hence fully faithful (Example 4.6.2.2). $\square$