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7.1.3 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 $y \in \mathscr {F}(Y)$ 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 7.1.3.1. 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 7.1.3.2. 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 7.1.3.3. 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 7.1.3.4. 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 7.1.3.5. 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 7.1.2.17, 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 4.4.2.11, we see that $\widetilde{e}$ is also an isomorphism, so that $\widetilde{X}$ is also a final object of $\widetilde{\operatorname{\mathcal{C}}}$ (Corollary 7.1.2.17). It follows that $X = U( \widetilde{X} )$ represents the right fibration $U$. $\square$

Proposition 7.1.3.5 admits the following converse:

Proposition 7.1.3.6. 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 5.1.6.1.

Corollary 7.1.3.7. 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 7.1.3.6 is an immediate consequence of a more precise assertion (Corollary 7.1.3.11) which we prove below. First, we need some preliminaries.

Proposition 7.1.3.8. 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 $\operatorname{\mathcal{C}}_{/Y}$.

$(2')$

The morphism $f$ is final when regarded as an object of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $.

$(3)$

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

$(3')$

The morphism $f$ is initial when regarded as an object of the oriented fiber product $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$.

Proof. The equivalences $(2) \Leftrightarrow (2')$ and $(3) \Leftrightarrow (3')$ follow from Corollaries 4.6.4.17 and 7.1.2.22. We will complete the proof by showing that $(1) \Leftrightarrow (3)$; the equivalence $(1) \Leftrightarrow (2)$ follows by applying the same argument in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. By virtue of Corollary 7.1.2.15, 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{equation} \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} \end{equation}

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 4.3.6.14, 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 $(1) \Leftrightarrow (3)$ now follows from the criterion of Theorem 4.4.2.6. $\square$

Corollary 7.1.3.9. 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 7.1.2.12), 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 7.1.3.8). $\square$

Corollary 7.1.3.10. 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 7.1.3.9. 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 7.1.3.11. 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 5.1.6.5). Since $\operatorname{id}_{Y}: Y \rightarrow Y$ is final when regarded as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$ (Proposition 7.1.3.8), Corollary 7.1.2.22 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 7.1.3.10 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. $\square$

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 7.1.3.12. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories and let $Y \in \operatorname{\mathcal{C}}$ be an object. Then:

$(1)$

Let $\widetilde{Y} \in \widetilde{\operatorname{\mathcal{C}}}$ be an object satisfying $U( \widetilde{Y} ) = Y$. Then $\widetilde{Y}$ is a final object of $\widetilde{\operatorname{\mathcal{C}}}$ if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\theta } \operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}_{Y}, \widetilde{\operatorname{\mathcal{C}}}_{X} ) \xrightarrow { \operatorname{ev}_{\widetilde{Y}} } \widetilde{\operatorname{\mathcal{C}}}_{X} \]

is a homotopy equivalence, where $\theta $ is given by parametrized contravariant transport (see Variant 5.2.7.6).

$(2)$

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 the homotopy transport representation of $U$, which we regard as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor (Variant 5.2.7.12). The object $Y$ represents the right fibration $U$ (in the sense of (in the sense of Definition 7.1.3.1) 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).

$(3)$

The right fibration $U$ is representable if and only if the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\operatorname{hTr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}}$ is representable.

Proof. Let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $ be the oriented fiber product of Construction 4.6.4.1, and let us regard $\operatorname{id}_{Y}$ as a final object of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $ (Proposition 7.1.3.8). Using Corollary 7.1.3.11, we can choose a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$ satisfying $F( \operatorname{id}_ Y ) = \widetilde{Y}$ which fits into a commutative diagram

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

Using Corollary 7.1.3.11, we see that $\widetilde{Y}$ is a final object of $\widetilde{\operatorname{\mathcal{C}}}$ if and only if $F$ is an equivalence of right fibrations over $\operatorname{\mathcal{C}}$. By virtue of Corollary 5.1.6.15, this is equivalent to the requirement that for each object $X \in \operatorname{\mathcal{C}}$, the functor $F$ restricts to a homotopy equivalence of Kan complexes

\[ F_{X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) = \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \{ X\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} = \widetilde{\operatorname{\mathcal{C}}}_{X} \]

Assertion $(1)$ follows from the observation that $F_{X}$ is homotopic to the composition of the parametrized contravariant transport morphism $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}_{Y}, \widetilde{\operatorname{\mathcal{C}}}_{X} )$ with the evaluation map $\operatorname{ev}_{ \widetilde{Y} }: \operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}_{Y}, \widetilde{\operatorname{\mathcal{C}}}_{X} ) \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{X}$ (see Remark 5.2.7.5 and Proposition 5.2.7.7). The implications $(1) \Rightarrow (2) \Rightarrow (3)$ are immediate from the definitions. $\square$

Corollary 7.1.3.13. 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 5.2.7.13 and Remark 5.2.7.5, 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 7.1.3.12. The implication $(3) \Rightarrow (2)$ is immediate, and the reverse implication follows from Proposition 7.1.3.6. $\square$

Corollary 7.1.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, and suppose that the object $X$ belongs to a full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $f$ exhibits $X$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$ (see Definition 6.2.2.1).

$(2)$

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

In particular, an object $X \in \operatorname{\mathcal{C}}'$ is a $\operatorname{\mathcal{C}}'$-reflection of $Y \in \operatorname{\mathcal{C}}$ if and only if it represents the right fibration $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}'$.

Proof. Let us regard the object $Y \in \operatorname{\mathcal{C}}$ as fixed, and let $\theta : \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}'$ be the right fibration given by projection onto the first factor. Using Example 5.2.7.13, we can identify the enriched homotopy transport representation of $\theta $ with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $(X \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}'}) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. The desired result now follows from the criterion of Proposition 7.1.3.12. $\square$