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8.2.6 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 8.2.6.1 (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 8.2.3.12). 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 8.2.5.4, 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 8.2.6.1. $\square$

Definition 8.2.6.2. 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 8.2.3.10, this condition does not depend on the choice of $\operatorname{Hom}$-functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, - )$.

Example 8.2.6.3. 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 5.6.6.2) 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 8.2.6.2).

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

Variant 8.2.6.4. 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 8.2.3.10, this condition does not depend on the choice of $\operatorname{Hom}$-functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, - )$.

Example 8.2.6.5. 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 8.2.6.2).

  • The profunctor $\mathscr {H}$ is corepresentable by the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (Variant 8.2.6.4).

Remark 8.2.6.6 (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 8.2.6.1 guarantees that $\mathscr {K}$ is representable (in the sense of Definition 8.2.2.4 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.

For many applications, Definition 8.2.6.2 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.

Definition 8.2.6.7. 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 $Y \in \operatorname{\mathcal{D}}$, the evaluation of $\beta $ at the object $\operatorname{id}_{Y} \in \operatorname{Tw}( \operatorname{\mathcal{D}})$ determines a vertex $\beta ( \operatorname{id}_{Y} ) \in \mathscr {K}( G(Y), Y )$ which exhibits the functor $\mathscr {K}( -, Y)$ as represented by the object $G(Y) \in \operatorname{\mathcal{C}}$ (see Variant 5.6.6.2).

Our main goal is to show that Definitions 8.2.6.2 and 8.2.6.7 are compatible. More precisely, we will prove at the end of this section that a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is representable by a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ if and only if there exists a natural transformation $\beta : \underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$ which satisfies the requirement of Definition 8.2.6.7 (see Proposition 8.2.6.15).

Example 8.2.6.8. In the situation of Definition 8.2.6.7, 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 8.2.6.7) if and only if it exhibits the functor $K$ as represented by the object $X$ (in the sense of Variant 5.6.6.2.

Example 8.2.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor. Then a natural transformation $\beta : \underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ exhibits $\mathscr {H}$ as represented by the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 8.2.6.7) if and only if it exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. This is a reformulation of Proposition 8.2.5.5.

Remark 8.2.6.10 (Homotopy Invariance). In the situation of Definition 8.2.6.7, 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 5.6.6.3).

Remark 8.2.6.11 (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 5.6.6.4.

Proposition 8.2.6.12. Suppose we are given a functor of $\infty $-categories $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$. Let $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$ be a natural transformation, which we identify with a functor of $\infty $-categories

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

Then $\beta $ exhibits $\mathscr {K}$ as represented by $G$ (in the sense of Definition 8.2.6.7) if and only if the functor $T$ is left cofinal.

Proof. We have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [dr]_-{\lambda _{+}} \ar [rr]^{T} & & \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}) \ar [dl]^-{\pi } \\ & \operatorname{\mathcal{D}}, & } \]

where the vertical maps are cocartesian fibrations and the functor $T$ carries $\lambda _{+}$-cocartesian morphisms of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ to $\pi $-cocartesian morphisms of the $\infty $-category $\{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}})$ (see Corollary 8.1.1.12). By virtue of Corollary 7.2.3.15, the functor $T$ is left cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the induced map

\[ T_{X}: \operatorname{Tw}( \operatorname{\mathcal{D}}) \times _{ \operatorname{\mathcal{D}}} \{ X\} \rightarrow \{ \Delta ^{0} \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{C}}^{\operatorname{op}} \]

is left cofinal; here the oriented fiber product on the right is formed with respect to the functor $\mathscr {K}(-, X): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$. Proposition 8.1.2.1 guarantees that the identity morphism $\operatorname{id}_{X}$ is initial when viewed as an object of the $\infty $-category $\infty $-category $\operatorname{Tw}( \operatorname{\mathcal{D}}) \times _{ \operatorname{\mathcal{D}}} \{ X\} $: that is, so that the inclusion map $\{ \operatorname{id}_ X \} \hookrightarrow \operatorname{Tw}( \operatorname{\mathcal{D}}) \times _{ \operatorname{\mathcal{D}}} \{ X\} $ is left cofinal (Example 7.2.1.4). Using Proposition 7.2.1.6, we see that $T_{X}$ is left cofinal if and only if the inclusion map $\{ \beta ( \operatorname{id}_ X ) \} \hookrightarrow \{ \Delta ^{0} \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{C}}^{\operatorname{op}}$ is left cofinal: that is, if and only if $\beta ( \operatorname{id}_ X )$ is initial when viewed as an object of the $\infty $-category $\{ \Delta ^{0} \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{C}}^{\operatorname{op}}$. By virtue of Proposition 5.6.6.22, this is equivalent to the requirement that $\beta ( \operatorname{id}_{X} )$ exhibits the functor $\mathscr {K}(-,X)$ as represented by the object $G(X) \in \operatorname{\mathcal{C}}$. $\square$

Proposition 8.2.6.13 (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 8.2.6.12). $\square$

Remark 8.2.6.14. Let $F: \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors 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 $\mathscr {K}'$ denote the composition

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}' \xrightarrow {\operatorname{id}\times F} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { \mathscr {K} } \operatorname{\mathcal{S}}, \]

which we regard as a profunctor from $\operatorname{\mathcal{D}}'$ to $\operatorname{\mathcal{C}}$. If $\beta : \underline{ \Delta ^0 } \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}})}$ is a natural transformation which exhibits $\mathscr {K}$ as represented by $G$, then the restriction $\beta |_{ \operatorname{Tw}( \operatorname{\mathcal{D}}' )}$ exhibits $\mathscr {K}'$ as represented by $G \circ F$.

Proposition 8.2.6.15. 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 Definitions 8.2.6.2.

$(2)$

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 8.2.6.7.

$(3)$

The functor $\mathscr {K}$ is a left Kan extension of the constant diagram $\underline{ \Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{D}})}$ along the composite functor

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

Proof. We first show that $(1)$ implies $(2)$. Since $\operatorname{\mathcal{C}}$ is locally small, it admits a $\operatorname{Hom}$-functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ (Proposition 8.2.3.10). Choose a natural transformation $\alpha : \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{C}})} \rightarrow \mathscr {H}$ which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. Then $\alpha $ exhibits the profunctor $\mathscr {H}$ as represented by the identity functor $\operatorname{id}_{ \operatorname{\mathcal{C}}}$ (Example 8.2.6.9). By virtue of Remark 8.2.6.11, we may assume that the profunctor $\mathscr {K}$ is given by the composition

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow {\operatorname{id}\times G} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\xrightarrow {\mathscr {H}} \operatorname{\mathcal{S}}\quad \quad (X,Y) \mapsto \mathscr {H}(X, G(Y) ). \]

Applying Remark 8.2.6.14, we see that precomposition with the functor $\operatorname{Tw}(G): \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{Tw}( \operatorname{\mathcal{C}})$ carries $\alpha $ to 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 $G$.

The implication $(2) \Rightarrow (3)$ follows from Proposition 8.2.6.13. It follows that $(1)$ implies $(3)$, and the reverse implication follows from the uniqueness of Kan extensions up to isomorphism (Remark 7.3.6.6). $\square$

Remark 8.2.6.16 (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 $G$. Combining Propositions 8.2.6.13 and 7.3.6.1, we see that precomposition with $\beta $ induces a homotopy equivalence

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

Example 8.2.6.17 (Spaces of Natural Transformation). Let $G,G': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ admits a $\operatorname{Hom}$-functor $\mathscr {H}$. Combining Remark 8.2.6.16 with Proposition 8.2.6.1, 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*}

In other words, the space of natural transformations from $G$ to $G'$ can be viewed as a limit of the diagram

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

For later use, we record a dual version of Definition 8.2.6.7.

Variant 8.2.6.18 (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 5.6.6.1. 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 8.2.2.3).

We say that a profunctor $\mathscr {K}$ is corepresentable by $F$ if there exists a natural transformation which exhibits $\mathscr {K}$ as corepresented by $F$. If the $\infty $-category $\operatorname{\mathcal{D}}$ admits a $\operatorname{Hom}$-functor $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(- , -): \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$, this condition is equivalent to the requirement that $\mathscr {K}$ is isomorphic to the profunctor given by the composite map

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