Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

8.3.4 Representable Profunctors

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. There is a fully faithful embedding from the category of functors $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ to the category of profunctors $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{Set})$, which assigns to each functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ the representable profunctor

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{Set}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) ). \]

This construction has an $\infty $-categorical counterpart:

Proposition 8.3.4.1 (Classification of Representable Profunctors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is locally small, and let

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, -): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}} \]

be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ (see Notation 8.3.3.7). Then the construction $G \mapsto \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( -, G(-) )$ determines a fully faithful functor

\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}), \]

whose essential image is spanned by the representable profunctors from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$.

Proof. Let $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ spanned by the representable functors. By virtue of Theorem 8.3.3.13, the construction $Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, Y)$ determines an equivalence of $\infty $-categories $h^{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. It follows that postcomposition with $h^{\bullet }$ induces an equivalence of $\infty $-categories

\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})), \]

which is a restatement of Proposition 8.3.4.1. $\square$

Definition 8.3.4.2. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is locally small and let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(-, -): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. We say that a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is representable by $G$ if it isomorphic to the composition

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow {\operatorname{id}\times G} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(-, -) } \operatorname{\mathcal{S}}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) ) \]

as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$. By virtue of Proposition 8.3.3.2, this condition does not depend on the choice of $\operatorname{Hom}$-functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, - )$.

Example 8.3.4.3. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category, and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be a functor. Then $\mathscr {F}$ is representable by an object $X \in \operatorname{\mathcal{C}}$ (in the sense of Variant 5.7.6.2) if and only if, when regarded as a profunctor from $\Delta ^0$ to $\operatorname{\mathcal{C}}$, it is representable by the functor $\Delta ^0 \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 8.3.4.2).

Interchanging the roles of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, we obtain the following dual notion:

Variant 8.3.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{D}}$ is locally small and let $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(-, -): \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{D}}$. We say that a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $F$ if it isomorphic to the composition

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow {F^{\operatorname{op}} \times \operatorname{id}} \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{D}}}(-, -) } \operatorname{\mathcal{S}}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), Y) \]

as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$. By virtue of Proposition 8.3.3.2, this condition does not depend on the choice of $\operatorname{Hom}$-functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, - )$.

Example 8.3.4.5. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category and let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{C}}$ to itself. The following conditions are equivalent:

  • The profunctor $\mathscr {H}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$.

  • The profunctor $\mathscr {H}$ is representable by the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (Definition 8.3.4.2).

  • The profunctor $\mathscr {H}$ is corepresentable by the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (Variant 8.3.4.4).

Remark 8.3.4.6. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories, $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $\lambda $. Then the profunctor $\mathscr {K}$ is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ (in the sense of Definition 8.3.4.2) if and only if the coupling $\lambda $ is representable by $G$ (in the sense of Definition 8.2.3.1). Similarly, $\mathscr {K}$ is corepresentable by a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ if and only if $\lambda $ is corepresentable by $F$.

Remark 8.3.4.7 (Uniqueness). Let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor of $\infty $-categories. If $\operatorname{\mathcal{C}}$ is locally small, then Proposition 8.3.4.1 guarantees that $\mathscr {K}$ is representable (in the sense of Definition 8.3.2.9 if and only if it is representable by $G$, for some functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Moreover, if this condition is satisfied, then the functor $G$ is determined uniquely to up isomorphism. Similarly, if $\operatorname{\mathcal{D}}$ is locally small, then $\mathscr {K}$ is corepresentable if and only if it is corepresentable by some functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. In this case, the functor $F$ is also uniquely determined up to isomorphism.

Example 8.3.4.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be locally small $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

  • The profunctor $\mathscr {K}$ is balanced (Definition 8.3.2.18).

  • The profunctor $\mathscr {K}$ is representable by a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ which is an equivalence of $\infty $-categories.

  • The profunctor $\mathscr {K}$ is corepresentable by a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is an equivalence of $\infty $-categories.

By virtue of Theorem 8.3.3.13, this is a reformulation of Corollary 8.3.2.20.

Proposition 8.3.4.9 (Adjunctions as Profunctors). Let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor of $\infty $-categories which represents a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$. Then a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ is left adjoint to $G$ if and only if it corepresents the profunctor $\mathscr {K}$. In particular, $\mathscr {K}$ is corepresentable if and only if the functor $G$ admits a left adjoint.

Proof. Choose a realization of $\mathscr {K}$ as the covariant transport representation of a coupling of $\infty $-categories $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ (see Remark 8.3.2.5). By virtue of Remark 8.3.4.6, the coupling $\lambda $ is representable by the functor $G$. By virtue of Theorem 8.2.5.1, a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ is left adjoint to $G$ if and only if it corepresents the coupling $\lambda $. Invoking Remark 8.3.4.6 again, we see that this is equivalent to the requirement that $F$ corepresents the profunctor $\mathscr {K}$. $\square$

For many applications, Definition 8.3.4.2 is insufficiently precise. Given a functor of $\infty $-categories $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, we would like to be able to consider not only profunctors which are representable by $G$ (meaning that they are abstractly isomorphic to the profunctor $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, G(Y) )$) but profunctors which are represented by $G$ (meaning that we have chosen an isomorphism with the profunctor $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,G(Y) )$, or some essentially equivalent datum). Here it is inconvenient that the functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(-,-)$ is well-defined only up to isomorphism. To address this point, it is convenient to encode representability in a different way.

Notation 8.3.4.10. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.6.1.1). We will regard the contractible Kan complex $\Delta ^0$ as an object of $\operatorname{\mathcal{S}}$. For every $\infty $-category $\operatorname{\mathcal{E}}$, we let $\underline{ \Delta ^0}_{\operatorname{\mathcal{E}}}$ denote the constant functor $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$ taking the value $\Delta ^0$.

Definition 8.3.4.11. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$, and let $\mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$ denote the composite functor

\[ \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { G^{\operatorname{op}} \times \operatorname{id}} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { \mathscr {K} } \operatorname{\mathcal{S}}. \]

Suppose we are given a natural transformation $\beta : \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) } \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$, where $\underline{\Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$ denotes the constant functor $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}}$ taking the value $\Delta ^0$. We say that $\beta $ exhibits the profunctor $\mathscr {K}$ as represented by $G$ if, for every object $D \in \operatorname{\mathcal{D}}$, the evaluation of $\beta $ at the object $\operatorname{id}_{D} \in \operatorname{Tw}( \operatorname{\mathcal{D}})$ determines a vertex $\beta ( \operatorname{id}_{D} ) \in \mathscr {K}( G(D), D )$ which exhibits the functor $\mathscr {K}( -, D)$ as represented by the object $G(D) \in \operatorname{\mathcal{C}}$ (see Variant 5.7.6.2).

Remark 8.3.4.12. In the situation of Definition 8.3.4.11, the natural transformation $\beta $ can be identified with a functor $\widetilde{G}$ which fits into a commutative diagram

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [r]^-{ \widetilde{G} } \ar [d] & \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}) \ar [d]^{\lambda } \\ \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. } \]

Moreover, the natural transformation $\beta $ exhibits $\mathscr {K}$ as represented by $G$ (in the sense of Definition 8.3.4.11) if and only if $\widetilde{G}$ exhibits the coupling $\lambda $ as represented by $G$ (in the sense of Definition 8.2.4.1).

Example 8.3.4.13. In the situation of Definition 8.3.4.11, suppose that $\operatorname{\mathcal{D}}= \Delta ^0$. In this case, we can identify the profunctor $\mathscr {K}$ with a functor $K: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$, we can identify the functor $G$ with an object $X \in \operatorname{\mathcal{C}}$, and we can identify $\beta $ with a vertex of the Kan complex $K(X)$. Then $\beta $ exhibits the profunctor $\mathscr {K}$ as represented by the functor $G$ (in the sense of Definition 8.3.4.11) if and only if it exhibits the functor $K$ as represented by the object $X$ (in the sense of Variant 5.7.6.2.

Proposition 8.3.4.14. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is locally small, and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. The following conditions are equivalent:

$(1)$

The profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is representable by $G$, in the sense of Definition 8.3.4.2.

$(2)$

There exists a natural transformation $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ which exhibits $\mathscr {K}$ as represented by $G$, in the sense of Definition 8.3.4.11.

Proof. By virtue of Remarks 8.3.4.6 and 8.3.4.12, this follows by applying Proposition 8.2.4.3 to the coupling

\[ \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. \]
$\square$

Variant 8.3.4.15 (Corepresentable Profunctors). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. We say that a natural transformation $\beta : \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{C}}) } \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{C}})}$ exhibits $\mathscr {K}$ as corepresented by $F$ if, for every object $X \in \operatorname{\mathcal{C}}$, the image $\beta ( \operatorname{id}_{X} ) \in \mathscr {K}( X, F(X) )$ exhibits the functor $\mathscr {K}( X, - ): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by the object $F(X) \in \operatorname{\mathcal{D}}$, in the sense of Definition 5.7.6.1. Equivalently, $\beta $ exhibits $\mathscr {K}$ as corepresented by $F$ if it exhibits $\mathscr {K}$ as represented by the opposite functor $F^{\operatorname{op}}$, when regarded as a profunctor from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to $\operatorname{\mathcal{D}}^{\operatorname{op}}$ (see Remark 8.3.2.3).

Remark 8.3.4.16 (Homotopy Invariance). In the situation of Definition 8.3.4.11, the condition that $\beta $ exhibits $\mathscr {K}$ as corepresented by $G$ depends only on the homotopy class $[\beta ]$ (as a morphism in the homotopy category $\mathrm{h} \mathit{ \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{S}})}$) (see Remark 5.7.6.3).

Remark 8.3.4.17 (Change of $\mathscr {K}$). Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Suppose we are given a pair of profunctors $\mathscr {K}, \mathscr {K}': \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$, a natural transformation $\alpha : \mathscr {K} \rightarrow \mathscr {K}'$, and a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & \underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} \ar [dl]^-{\beta } \ar [dr]_-{\beta '} & \\ \mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} \ar [rr]^{ \alpha |_{ \operatorname{Tw}( \operatorname{\mathcal{D}})} } & & \mathscr {K}'|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} } \]

in the $\infty $-category $\operatorname{Fun}( \operatorname{Tw}( \operatorname{\mathcal{D}}), \operatorname{\mathcal{S}})$. Then any two of the following conditions imply the third:

  • The natural transformation $\beta $ exhibits the profunctor $\mathscr {K}$ as represented by $G$.

  • The natural transformation $\beta '$ exhibits the profunctor $\mathscr {K}'$ as represented by $G$.

  • The natural transformation $\alpha $ is an isomorphism.

See Remark 5.7.6.4.

Proposition 8.3.4.18. Suppose we are given a functor of $\infty $-categories $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$, and a natural transformation $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$. Then $\beta $ exhibits $\mathscr {K}$ as represented by $G$ (in the sense of Definition 8.3.4.11) if and only if the induced map $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}})$ is left cofinal.

Proof. By virtue of Remark 8.3.4.12, this is a special case of Proposition 8.2.4.9. $\square$

Proposition 8.3.4.19 (Representable Profunctors as Kan Extensions). Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor, and let $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ be a natural transformation which exhibits $\mathscr {K}$ as represented by $G$. Then $\beta $ exhibits $\mathscr {K}$ as a left Kan extension of the constant diagram $\underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ along the composite map

\[ \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { G^{\operatorname{op}} \times \operatorname{id}} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. \]

Proof. Let $\operatorname{\mathcal{E}}$ denote the oriented fiber product $\{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}})$ and let $\mu : \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$ be the projection onto the second factor, so that we have a tautological natural transformation $\widetilde{\beta }: \underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {K} \circ \mu $. It follows from Proposition 7.6.2.15 that $\widetilde{\beta }$ exhibits $\mathscr {K}$ as a left Kan extension of $\underline{\Delta ^0}_{\operatorname{\mathcal{E}}}$ along $\mu $. The natural transformation $\beta $ then determines a functor $T: \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{E}}$ such that precomposition with $T$ carries $\widetilde{\beta }$ to $\beta $. By the transitivity of the formation of of Kan extensions (Proposition 7.3.7.18), we are reduced to showing that the identity transformation $\operatorname{id}: \underline{ \Delta ^0}_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) } \rightarrow \underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}} \circ T$ exhibits $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$ as a left Kan extension of $\underline{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ along $T$. This is a special case of Remark 7.6.2.12, since the functor $T$ is left cofinal (Proposition 8.3.4.18). $\square$

Corollary 8.3.4.20 (The Universal Mapping Property of Representable Profunctors). Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Suppose we are given a pair of profunctors $\mathscr {K}, \mathscr {K}': \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$, and let $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ be a natural transformation which exhibits $\mathscr {K}$ as represented by $G$. Then precomposition with $\beta $ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) }( \mathscr {K}, \mathscr {K}') \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{Tw}( \operatorname{\mathcal{D}}), \operatorname{\mathcal{S}})}( \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{D}})}, \mathscr {K}'|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} ). \]

Example 8.3.4.21 (Spaces of Natural Transformation). Let $G,G': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $\mathscr {H}$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. Combining Corollary 8.3.4.20 with Proposition 8.3.4.1, we obtain homotopy equivalences of Kan complexes

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})}( G, G' ) & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) }( \mathscr {H} \circ (\operatorname{id}\times G), \mathscr {H} \circ (\operatorname{id}\times G') ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{S}})}( \underline{\Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}, \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}}) }) \\ & \simeq & \varprojlim ( \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}}) }). \end{eqnarray*}

Stated more informally, the space of natural transformations from $G$ to $G'$ can be viewed as a limit of the diagram

\[ \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}}\quad \quad (f: X \rightarrow Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(X), G'(Y)). \]

Corollary 8.3.4.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms in $\operatorname{\mathcal{C}}$. Let $K$ be a simplicial set, and let $\widetilde{W}$ be the collection of morphisms in $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ consisting of those natural transformations $w: F \rightarrow F'$ with the property that, for each vertex $x \in K$, the image $w_{x}: F(x) \rightarrow F'(x)$ belongs to $W$. If $G: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram which carries each vertex of $K$ to a $W$-local object of $\operatorname{\mathcal{C}}$, then $G$ is $\widetilde{W}$-local when viewed as an object of $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ (see Definition 6.3.3.1)

Proof. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $K \hookrightarrow \operatorname{\mathcal{K}}$, where $\operatorname{\mathcal{K}}$ is an $\infty $-category. Extend $G$ to a functor $\overline{G}: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$. Since the restriction map $\operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is a trivial Kan fibration (Proposition 1.4.7.6), we can replace $K$ by $\operatorname{\mathcal{K}}$ (and $G$ by $\overline{G}$) and thereby reduce to proving Corollary 8.3.4.22 in the special case where $K = \operatorname{\mathcal{K}}$ is an $\infty $-category.

Let $w: F \rightarrow F'$ be a morphism in $\operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})$ which belongs to $\widetilde{W}$; we wish to show that precomposition with $w$ induces a homotopy equivalence of Kan complexes $\theta : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) }(F',G) \xrightarrow { \circ [w] } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) }( F, G )$. Let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, let $U: \operatorname{Tw}(\operatorname{\mathcal{K}}) \rightarrow \operatorname{\mathcal{S}}$ be the functor given by the composition

\[ \operatorname{Tw}(\operatorname{\mathcal{K}}) \rightarrow \operatorname{\mathcal{K}}^{\operatorname{op}} \times \operatorname{\mathcal{K}}\xrightarrow {F^{\operatorname{op}} \times G} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\xrightarrow { \mathscr {H}} \operatorname{\mathcal{S}} \]

and define $U': \operatorname{Tw}(\operatorname{\mathcal{K}}) \rightarrow \operatorname{\mathcal{S}}$ similarly. The morphism $w$ induces a natural transformation $\overline{w}: U' \rightarrow U$, which carries each object $(f: X \rightarrow Y)$ of $\operatorname{Tw}(\operatorname{\mathcal{K}})$ to the morphism of Kan complexes

\[ \overline{w}_{f}: U'(f) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F'(X), G(Y) ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(X), G(Y) ) \simeq U(f) \]

given by precomposition with the morphism $w_{X}: F'(X) \rightarrow F(X)$ of $\operatorname{\mathcal{C}}$. Since $w_{X}$ belongs to $W$ and the object $G(Y) \in \operatorname{\mathcal{C}}$ is $W$-local, it follows that $\overline{w}_{f}$ is a homotopy equivalence. Applying Theorem 4.4.4.4, we see that $\overline{w}$ is an isomorphism (in the functor $\infty $-category $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{K}}), \operatorname{\mathcal{S}})$). We conclude by observing that Example 8.3.4.21 supplies an identification of $\theta $ with the image of $\overline{w}$ under the limit functor $\varprojlim : \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{K}}), \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$. $\square$