# Kerodon

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### 7.1.2 Representable Fibrations

Let $\operatorname{\mathcal{C}}$ be a category. Recall that a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is representable if there exists an object $Y \in \operatorname{\mathcal{C}}$ and an element $y \in \mathscr {F}(C)$ with the following universal property: for every object $X \in \operatorname{\mathcal{C}}$, evaluation on $y$ induces a bijection $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \mathscr {F}(X)$. Unwinding the definitions, we see that this is equivalent to the requirement that the pair $(Y,y)$ is a final object of the category of elements $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ introduced in Variant 5.5.1.2. This motivates the following:

Definition 7.1.2.1. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty$-categories. We say that $\widetilde{\operatorname{\mathcal{C}}}$ is represented by an object $Y \in \operatorname{\mathcal{C}}$ if there exists a final object $\widetilde{Y} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $U( \widetilde{Y} ) = Y$. In this case, we say that $\widetilde{Y}$ exhibits $\widetilde{\operatorname{\mathcal{C}}}$ as a right fibration represented by $Y$. We say that a right fibration $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is representable if it it is represented by some object of $\operatorname{\mathcal{C}}$: that is, if the $\infty$-category $\widetilde{\operatorname{\mathcal{C}}}$ has a final object.

Variant 7.1.2.2. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty$-categories. We say that $\widetilde{\operatorname{\mathcal{C}}}$ is corepresented by an object $X \in \operatorname{\mathcal{C}}$ if there exists an initial object $\widetilde{X} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $U( \widetilde{X} ) = X$. In this case, we say that $\widetilde{X}$ exhibits $\widetilde{\operatorname{\mathcal{C}}}$ as a left fibration corepresented by $X$. We say that a left fibration $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is corepresentable if it it is corepresented by some object of $\operatorname{\mathcal{C}}$: that is, if the $\infty$-category $\widetilde{\operatorname{\mathcal{C}}}$ has an initial object.

Example 7.1.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is both a left fibration and a right fibration. An object $Y \in \operatorname{\mathcal{C}}$ represents the right fibration $\operatorname{id}_{\operatorname{\mathcal{C}}}$ if and only if $Y$ is a final object of $\operatorname{\mathcal{C}}$, and corepresents the left fibration $\operatorname{id}_{\operatorname{\mathcal{C}}}$ if and only if $Y$ is an initial object of $\operatorname{\mathcal{C}}$.

Remark 7.1.2.4. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty$-categories. Then $U$ is represented by an object $Y \in \operatorname{\mathcal{C}}$ if and only if the left fibration $U^{\operatorname{op}}: \widetilde{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is corepresented by $Y$.

If $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is a representable right fibration, then the representing object is uniquely determined up to isomorphism:

Proposition 7.1.2.5. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty$-categories which is represented by an object $Y \in \operatorname{\mathcal{C}}$. Let $X$ be another object of $\operatorname{\mathcal{C}}$. Then $X$ represents the right fibration $U$ if and only if it is isomorphic to $Y$.

Proof. Since $U$ is represented by $Y$, there exists a final object $\widetilde{Y} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $U( \widetilde{Y} ) = Y$. If $U$ is also represented by $X$, then we can choose another final object $\widetilde{X} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $U( \widetilde{X} ) = X$. Applying Corollary 7.1.1.14, we deduce that there exists an isomorphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty$-category $\widetilde{\operatorname{\mathcal{C}}}$. Then $e = U( \widetilde{e} )$ is an isomorphism from $X$ to $Y$ in the $\infty$-category $\operatorname{\mathcal{C}}$.

For the converse, suppose that there exists an isomorphism $e: X \rightarrow Y$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Since $U$ is a right fibration, we can lift $e$ to a morphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty$-category $\widetilde{\operatorname{\mathcal{C}}}$. Applying Proposition 4.4.2.10, we see that $\widetilde{e}$ is also an isomorphism, so that $\widetilde{X}$ is also a final object of $\widetilde{\operatorname{\mathcal{C}}}$ (Corollary 7.1.1.14). It follows that $X = U( \widetilde{X} )$ represents the right fibration $U$. $\square$

Proposition 7.1.2.5 admits the following converse:

Proposition 7.1.2.6. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty$-categories. Then $U$ is represented by an object $Y \in \operatorname{\mathcal{C}}$ if and only if it is equivalent to the right fibration $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$, in the sense of Definition 5.6.2.1.

Corollary 7.1.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the construction $Y \mapsto \operatorname{\mathcal{C}}_{/Y}$ induces a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Objects of \operatorname{\mathcal{C}}} \} / \textnormal{Isomorphism} \ar [d] \\ \{ \textnormal{Representable right fibrations \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}} \} / \textnormal{Equivalence}.}$

Proposition 7.1.2.6 is an immediate consequence of a more precise assertion (Corollary 7.1.2.11) which we prove below. First, we need some preliminaries.

Proposition 7.1.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $f$ is an isomorphism from $X$ to $Y$ in the $\infty$-category $\operatorname{\mathcal{C}}$ (Definition 1.3.6.1).

$(2)$

The morphism $f$ is final when regarded as an object of the slice $\infty$-category $\operatorname{\mathcal{C}}_{/Y}$.

$(3)$

The morphism $f$ is initial when regarded as an object of the coslice $\infty$-category $\operatorname{\mathcal{C}}_{X/}$.

Proof. By virtue of Corollary 7.1.1.12, condition $(3)$ is equivalent to the requirement that the restriction map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{X/}$ is a trivial Kan fibration: that is, every lifting problem

7.3
$$\begin{gathered}\label{equation:limit-of-point} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{f/} \ar [d] \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}_{X/} } \end{gathered}$$

admits a solution. Using the isomorphism of simplicial sets

$(\Delta ^1 \star \operatorname{\partial \Delta }^ n) \coprod _{ \{ 0\} \star \operatorname{\partial \Delta }^ n } ( \{ 0\} \star \Delta ^ n ) \simeq \Lambda ^{n+2}_{0}$

supplied by Lemma 4.3.6.12, we can identify (7.3) with a lifting problem

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{n+2}_{0} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ \Delta ^{n+2}_{0} \ar@ {-->}[ur]^{\sigma } \ar [r] & \Delta ^0, }$

where $\sigma _0$ carries the initial edge $\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Lambda ^{n+2}_{0}$ to the morphism $f$. The equivalence of $(1)$ and $(3)$ now follows from the criterion of Theorem 4.4.2.6. The equivalence of $(1)$ and $(2)$ follows by applying the same argument to the opposite $\infty$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. $\square$

Corollary 7.1.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

The object $Y$ is final if and only if the projection map $F: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ admits a section $G'$ satisfying $G(Y) = \operatorname{id}_{Y}$.

$(2)$

The object $Y$ is initial if and only if the projection map $F': \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}$ admits a section $G'$ satisfying $G'(Y) = \operatorname{id}_{Y}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. If $Y$ is a final object, then the projection map $F: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration (Proposition 7.1.1.9), so the construction $Y \mapsto \operatorname{id}_ Y$ can be extended to a section of $F$. Conversely, suppose that $F$ admits a section $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{/Y}$ satisfying $G(Y) = \operatorname{id}_ Y$. Let $X$ be an object of $\operatorname{\mathcal{C}}$: we wish to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is contractible. The functors $G$ and $F$ induce morphisms of Kan complexes

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/Y}}( G(X), \operatorname{id}_ Y ) \xrightarrow {F} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y),$

whose composition is the identity. In particular, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a retract of $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{/Y}}( G(X), \operatorname{id}_ Y )$. It will therefore suffice to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{/Y}}( G(X), \operatorname{id}_ Y )$ is contractible. This is clear, since $\operatorname{id}_{Y}$ is a final object of the slice $\infty$-category $\operatorname{\mathcal{C}}_{/Y}$ (Proposition 7.1.2.8). $\square$

Corollary 7.1.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The object $Y \in \operatorname{\mathcal{C}}$ is final.

$(2)$

There exists a functor $F: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $F|_{\operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and for which the composition

$\Delta ^1 \simeq \{ Y\} ^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright } \xrightarrow {F} \operatorname{\mathcal{C}}$

is the identity morphism $\operatorname{id}_{Y}$ (in particular, $F$ carries the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$ to the object $Y$).

$(3)$

The inclusion map $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is right anodyne.

Proof. The equivalence $(1) \Leftrightarrow (2)$ is a reformulation of Corollary 7.1.2.9. We next show that $(2)$ implies $(3)$. If condition $(2)$ is satisfied, then we have a commutative diagram of simplicial sets

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

where the horizontal compositions are the identity. Since the inclusion $\{ Y\} ^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright }$ is right anodyne (Lemma 4.3.7.8), it follows that the inclusion $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is also right anodyne.

We now complete the proof by showing that $(3)$ implies $(2)$. Suppose that the inclusion $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is right anodyne; we wish to show that there exists a functor $F: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $F|_{ \operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and $F|_{ \{ Y\} ^{\triangleright } } = \operatorname{id}_ Y$. For this, it will suffice to show that the inclusion map

$\operatorname{\mathcal{C}}\coprod _{ \{ Y\} } \{ Y\} ^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright }$

is inner anodyne, which is a special case of Proposition 4.3.6.4. $\square$

Corollary 7.1.2.11. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty$-categories and let $\widetilde{Y} \in \widetilde{\operatorname{\mathcal{C}}}$ be an object having image $Y = U( \widetilde{Y} )$. The following conditions are equivalent:

$(1)$

There exists an equivalence $F: \operatorname{\mathcal{C}}_{/Y} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$ of right fibrations over $\operatorname{\mathcal{C}}$ satisfying $F( \operatorname{id}_ Y) = \widetilde{Y}$.

$(2)$

The object $\widetilde{Y}$ exhibits $U$ as a right fibration represented by $Y$: that is, it is a final object of the $\infty$-category $\widetilde{\operatorname{\mathcal{C}}}$.

$(3)$

For every right fibration $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, evaluation on the object $\widetilde{Y}$ induces a trivial Kan fibration $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \widetilde{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \{ Y\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$.

$(4)$

For every right fibration $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, evaluation on the object $\widetilde{Y}$ induces a bijection

$\pi _0( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \widetilde{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) ) \rightarrow \pi _0( \{ Y\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}).$

Proof. If $F: \operatorname{\mathcal{C}}_{/Y} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$ is an equivalence of right fibrations over $\operatorname{\mathcal{C}}$, then it is an equivalence of $\infty$-categories (Proposition 5.6.2.5). Since $\operatorname{id}_{Y}: Y \rightarrow Y$ is final when regarded as an object of the $\infty$-category $\operatorname{\mathcal{C}}_{/Y}$ (Proposition 7.1.2.8), Corollary 7.1.1.19 guarantees that $\widetilde{Y}$ is a final object of $\widetilde{\operatorname{\mathcal{C}}}$. This proves the implication $(1) \Rightarrow (2)$. The implication $(2) \Rightarrow (3)$ follows by combining Corollary 7.1.2.10 with Proposition 4.2.5.4, and the implication $(3) \Rightarrow (4)$ is trivial.

We will complete the proof by showing that $(4)$ implies $(1)$. Note that the object $\operatorname{id}_{Y} \in \operatorname{\mathcal{C}}_{/Y}$ satisfies condition $(1)$ and therefore also satisfies condition $(3)$. It follows that there exists a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/Y} \ar [rr]^{F} \ar [dr] & & \widetilde{\operatorname{\mathcal{C}}} \ar [dl]^{U} \\ & \operatorname{\mathcal{C}}& }$

satisfying $F( \operatorname{id}_{Y} ) = \widetilde{Y}$. To complete the proof, it will suffice to show that $F$ is an equivalence of right fibrations over $\operatorname{\mathcal{C}}$. For every right fibration $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, we have a commutative diagram of sets

$\xymatrix@R =50pt@C=50pt{ \pi _0( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \widetilde{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) ) \ar [rr]^{\circ [F]} \ar [dr] & & \pi _0( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}_{/Y}, \operatorname{\mathcal{D}}) ) \ar [dl] \\ & \pi _0( \{ Y\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}) & , }$

where the vertical maps are given by evaluation on the objects $\widetilde{Y} \in \widetilde{\operatorname{\mathcal{C}}}$ and $\operatorname{id}_{Y} \in \operatorname{\mathcal{C}}_{/Y}$, and are therefore bijective. It follows that the horizontal map is also bijective. $\square$

Example 7.1.2.12. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty$-categories. Suppose we are given an object $\widetilde{Y}$ of $\widetilde{\operatorname{\mathcal{C}}}$ having image $Y \in \operatorname{\mathcal{C}}$. It follows from Corollary 7.1.2.11 that there exists a functor of $\infty$-categories $F: \operatorname{\mathcal{C}}_{/Y} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$ satisfying $F( \operatorname{id}_ Y ) = \widetilde{Y}$, which fits into a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/Y} \ar [rr]^{F} \ar [dr] & & \widetilde{\operatorname{\mathcal{C}}} \ar [dl]^{U} \\ & \operatorname{\mathcal{C}}; & }$

moreover, the collection of such functors is parametrized by a contractible Kan complex. Note that $F$ induces a natural transformation of homotopy transport functors

$\operatorname{hTr}_{F}: \operatorname{hTr}_{ \operatorname{\mathcal{C}}_{/Y} / \operatorname{\mathcal{C}}} \rightarrow \operatorname{hTr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}}$

(Variant 5.2.3.10), which is an isomorphism if and only if $\widetilde{Y}$ is a final object of $\widetilde{\operatorname{\mathcal{C}}}$ (Corollary 7.1.2.11).

Using Example 5.2.3.11, we can identify the value of $\operatorname{hTr}_{\operatorname{\mathcal{C}}_{/Y} / \operatorname{\mathcal{C}}}$ on an object $X \in \operatorname{\mathcal{C}}$ with the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. Under this identification, the value of the natural transformation $\operatorname{hTr}_{F}$ on $X$ is given by the map

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \{ \widetilde{Y} \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \widetilde{\operatorname{\mathcal{C}}}_{Y} \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{X}$

determined by the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment of the functor $\operatorname{hTr}_{\widetilde{\operatorname{\mathcal{C}}}/\operatorname{\mathcal{C}}}$ (see Variant 5.2.3.10).

The representability of a right fibration $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ can be tested at the level of its homotopy transport representation:

Proposition 7.1.2.13. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty$-categories, and let $\operatorname{hTr}_{\widetilde{\operatorname{\mathcal{C}}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ be its homotopy transport representation, which we regard as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor (Variant 5.2.3.10). Then an object $Y \in \operatorname{\mathcal{C}}$ represents the right fibration $U$ (in the sense of Definition 7.1.2.1) if and only if it represents the functor $\operatorname{hTr}_{\widetilde{\operatorname{\mathcal{C}}}/\operatorname{\mathcal{C}}}$ (in the sense of Definition 6.2.4.1). In particular, the right fibration $U$ is representable if and only if the functor $\operatorname{hTr}_{\widetilde{\operatorname{\mathcal{C}}}/\operatorname{\mathcal{C}}}$ is representable.

Proof. By definition, the object $Y$ represents the functor $\operatorname{hTr}_{\widetilde{\operatorname{\mathcal{C}}}/\operatorname{\mathcal{C}}}$ if and only if there exists an object

$\widetilde{Y} \in \operatorname{hTr}_{\widetilde{\operatorname{\mathcal{C}}}/\operatorname{\mathcal{C}}}(Y) = \{ Y\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}}$

with the following property: for every object $X \in \operatorname{\mathcal{C}}$, the induced map

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \{ \widetilde{Y} \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{hTr}_{\widetilde{\operatorname{\mathcal{C}}}/\operatorname{\mathcal{C}}}(Y) \rightarrow \operatorname{hTr}_{\widetilde{\operatorname{\mathcal{C}}}/\operatorname{\mathcal{C}}}(X)$

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. By virtue of Example 7.1.2.12, this is equivalent to the requirement that $\widetilde{Y}$ is a final object of $\widetilde{\operatorname{\mathcal{C}}}$. $\square$

Corollary 7.1.2.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. The following conditions are equivalent:

$(1)$

The functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

$(2)$

For every object $Y \in \operatorname{\mathcal{D}}$, the right fibration $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is representable.

$(3)$

For every representable right fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$, the right fibration $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{C}}$ is representable.

If these conditions are satisfied, then the functor $G$ carries each object $Y \in \operatorname{\mathcal{D}}$ to an object of $\operatorname{\mathcal{C}}$ which represents the right fibration $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$.

Proof. By virtue of Corollary 6.2.4.7, the functor $F$ admits a right adjoint if and only if, for every object $Y \in \operatorname{\mathcal{D}}$, the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

$\theta : \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad \theta (X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(X), Y)$

is representable. Using Example 5.2.3.11 and Remark 5.2.2.4, we can identify $\theta$ with the homotopy transport representation for the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ Y \} \rightarrow \operatorname{\mathcal{C}}$. The equivalence of $(1)$ and $(2)$ now follows from Proposition 7.1.2.13. The implication $(3) \Rightarrow (2)$ is immediate, and the reverse implication follows from Proposition 7.1.2.6. $\square$