Definition 8.3.2.1. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. A profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ is a functor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
8.3.2 Profunctors of $\infty $-Categories
Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be categories. A profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ is a $\operatorname{Set}$-valued functor on the product category $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$. This notion has an evident $\infty $-categorical analogue, where we replace the ordinary category of sets by the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces (see Construction 5.5.1.1).
Example 8.3.2.2. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be ordinary categories. Then every functor $K: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{Set}$ determines a morphism of simplicial sets This construction determines a monomorphism from the collection of profunctors from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ (in the sense of classical category theory) to the collection of profunctors from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{+})$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{-})$ (in the sense of Definition 8.3.2.1). Beware that this map is (usually) not bijective: its image consists of those profunctors having the property that for every pair of objects $X \in \operatorname{\mathcal{C}}_{-}$ and $Y \in \operatorname{\mathcal{C}}_{+}$, the Kan complex $\mathscr {K}(X,Y)$ is a constant simplicial set (see Proposition 1.3.3.1 and Remark 5.5.1.7).
\[ \operatorname{N}_{\bullet }(K): \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{-} )^{\operatorname{op}} \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{+} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{\mathcal{S}}. \]
\[ \mathscr {K}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{-})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{+}) \rightarrow \operatorname{\mathcal{S}} \]
Remark 8.3.2.3 (Symmetry). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$. Then, by transposing its arguments, we can also regard $\mathscr {K}$ as a profunctor from $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ to $\operatorname{\mathcal{C}}_{+}^{\operatorname{op}}$.
Example 8.3.2.4 (From Profunctors to Couplings). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ (Definition 8.3.2.1). Applying the construction of Definition 5.6.2.1, we obtain an $\infty $-category $\int _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} } \mathscr {K}$ whose objects are triples $(X,Y,\eta )$ where $X$ is an object of $\operatorname{\mathcal{C}}_{-}$, $Y$ is an object of $\operatorname{\mathcal{C}}_{+}$, and $\eta $ is a vertex of the Kan complex $\mathscr {K}(X,Y)$. This $\infty $-category is equipped with a left fibration given on objects by the construction $\lambda (X,Y,\eta ) = (X,Y)$ (see Example 5.6.2.9). The left fibration $\lambda $ is a coupling of $\operatorname{\mathcal{C}}_{+}$ with $\operatorname{\mathcal{C}}_{-}$, in the sense of Definition 8.2.0.1; we will refer to it as the coupling associated to the profunctor $\mathscr {K}$.
\[ \lambda : \int _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} } \mathscr {K} \rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}, \]
Modulo set-theoretic issues, every coupling can be obtained from the construction of Example 8.3.2.4:
Remark 8.3.2.5 (From Couplings to Profunctors). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. By virtue of Corollary 5.6.0.6, the construction of Example 8.3.2.4 induces a monomorphism whose image consists of equivalence classes of couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ having essentially small fibers. In particular, to every coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ with essentially small fibers, we can associate a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$, which is characterized (up to isomorphism) by the requirement that it is a covariant transport representation for $\lambda $ (see Definition 5.6.5.1).
\[ \xymatrix@C =50pt@R=50pt{ \{ \textnormal{Profunctors $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$} \} / \textnormal{Isomorphism} \ar [d] \\ \{ \textnormal{Couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$} \} / \textnormal{Equivalence}, } \]
Variant 8.3.2.6. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, and let $\kappa $ be an uncountable cardinal. Then the construction of Example 8.3.2.4 induces a monomorphism whose image consists of equivalence classes of couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ whose fibers are essentially $\kappa $-small.
\[ \xymatrix@C =50pt@R=50pt{ \{ \textnormal{Profunctors $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$} \} / \textnormal{Isomorphism} \ar [d] \\ \{ \textnormal{Couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$} \} / \textnormal{Equivalence}, } \]
Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. A profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ can be identified with a functor from $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ to the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}})$. Our primary goal in this section is to formulate a condition which guarantees that this functor is fully faithful. First, it will be convenient to introduce some terminology.
Definition 8.3.2.7. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor. Let $X$ be an object of $\operatorname{\mathcal{C}}_{-}$, let $Y$ be an object of $\operatorname{\mathcal{C}}_{+}$, and let $\eta $ be a vertex of the Kan complex $\mathscr {K}(X,Y)$. We will say that $\eta $ is universal if it exhibits the functor $\mathscr {K}(-,Y): \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by the object $X \in \operatorname{\mathcal{C}}_{-}$. We say that $\eta $ is couniversal if it exhibits the functor $\mathscr {K}(X, -): \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ as corepresented by the object $Y \in \operatorname{\mathcal{C}}_{+}$.
Remark 8.3.2.8. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $\lambda $. Let $C$ be an object of $\operatorname{\mathcal{C}}$ and set $\lambda (C) = (X,Y)$. Then the isomorphism class of $C$ (as an object of the fiber $\lambda ^{-1} \{ (X,Y ) \} $) can be identified with a connected component $[\eta ]$ of the Kan complex $\mathscr {K}(X,Y )$. Invoking Proposition 5.6.6.21, we deduce the following:
The object $C \in \operatorname{\mathcal{C}}$ is universal (in the sense of Definition 8.2.1.1) if and only if the vertex $\eta \in \mathscr {K}(X,Y)$ is universal (in the sense of Definition 8.3.2.7).
The object $C \in \operatorname{\mathcal{C}}$ is couniversal (in the sense of Definition 8.2.1.1) if and only if the vertex $\eta \in \mathscr {K}(X,Y)$ is couniversal (in the sense of Definition 8.3.2.7).
Definition 8.3.2.9. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$. We say that $\mathscr {K}$ is representable if, for each object $Y \in \operatorname{\mathcal{C}}_{+}$, the functor $\mathscr {K}(-, Y): \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable (in the sense of Variant 5.6.6.2). We will say that $\mathscr {K}$ is corepresentable if, for each object $X \in \operatorname{\mathcal{C}}_{-}$, the functor $\mathscr {K}(X, -): \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ is corepresentable (in the sense of Definition 5.6.6.1).
Warning 8.3.2.10. The terminology of Definition 8.3.2.9 is potentially confusing. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, let $\operatorname{\mathcal{C}}$ denote the product $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$, and let $\mathscr {K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a morphism of simplicial sets. In general, there is no relationship between the corepresentability of $\mathscr {K}$ as a $\operatorname{\mathcal{S}}$-valued functor on $\operatorname{\mathcal{C}}$ (in the sense of Definition 5.6.6.1) and the corepresentability of $\mathscr {K}$ as a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ (in the sense of Definition 8.3.2.9). However, these notions of corepresentability coincide when $\operatorname{\mathcal{C}}_{-}$ is a contractible Kan complex (see Example 8.3.2.13).
Remark 8.3.2.11 (Symmetry). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$. Then $\mathscr {K}$ is representable if and only if it is corepresentable when regarded as a profunctor from $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ to $\operatorname{\mathcal{C}}_{+}^{\operatorname{op}}$ (see Remark 8.3.2.3).
Remark 8.3.2.12. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}$ and $\mathscr {K}'$ be profunctors from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ which are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}})$). Then $\mathscr {K}$ is representable if and only if $\mathscr {K}'$ is representable. Similarly, $\mathscr {K}$ is corepresentable if and only if $\mathscr {K}'$ is corepresentable. See Remark 5.6.6.4.
Example 8.3.2.13. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor. If $\operatorname{\mathcal{C}}_{-} = \Delta ^0$, then the profunctor $\mathscr {K}$ is corepresentable (in the sense of Definition 8.3.2.9) if and only if it is corepresentable when regarded as a functor $\operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ (in the sense of Definition 5.6.6.1). Similarly, if $\operatorname{\mathcal{C}}_{+} = \Delta ^0$, then the profunctor $\mathscr {K}$ is representable (in the sense of Definition 8.3.2.9) if and only if it is representable when viewed as a functor $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ (in the sense of Variant 5.6.6.2).
Exercise 8.3.2.14. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be ordinary categories. Show that a profunctor is representable (in the sense of Definition 8.3.2.9) if and only if it it is isomorphic to the profunctor $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}_{-}}( X, G( Y) )$, for some functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$. See Proposition 8.3.4.1 for a more general result.
\[ \mathscr {K}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{-})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{+}) \rightarrow \operatorname{\mathcal{S}} \]
Remark 8.3.2.15. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor. Then $\mathscr {K}$ is representable if and only if, for every object $Y \in \operatorname{\mathcal{C}}_{+}$, there exists an object $X \in \operatorname{\mathcal{C}}_{-}$ and a universal vertex $\eta \in \mathscr {K}( X,Y)$. Similarly, $\mathscr {K}$ is corepresentable if and only if, for every object $X \in \operatorname{\mathcal{C}}_{-}$, there exists an object $Y \in \operatorname{\mathcal{C}}_{+}$ and a couniversal vertex $\eta \in \mathscr {K}(X,Y)$.
Remark 8.3.2.16. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $\lambda $. Using Remark 8.3.2.8, we deduce the following:
The coupling $\lambda $ is representable (in the sense of Definition 8.2.1.3) if and only if the profunctor $\mathscr {K}$ is representable (in the sense of Definition 8.3.2.9).
The coupling $\lambda $ is corepresentable (in the sense of Definition 8.2.1.3) if and only if the profunctor $\mathscr {K}$ is corepresentable (in the sense of Definition 8.3.2.9).
The main result of this section is the following variant of Proposition 8.2.6.8:
Proposition 8.3.2.17. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a corepresentable profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$. The following conditions are equivalent:
The profunctor $\mathscr {K}$ determines a fully faithful functor
Let $X$ be an object of $\operatorname{\mathcal{C}}_{-}$ and let $Y$ be an object of $\operatorname{\mathcal{C}}_{+}$. Then every couniversal vertex $\eta \in \mathscr {K}(X,Y)$ is also universal.
\[ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, -). \]
Proof. Choose an object $X \in \operatorname{\mathcal{C}}_{-}$. Since the functor $\mathscr {K}(X, -): \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ is corepresentable, we can choose an object $Y \in \operatorname{\mathcal{C}}_{+}$ and a couniversal vertex $\eta \in \mathscr {K}(X,Y)$. We will show that the following conditions are equivalent:
- $(1_ X)$
For every object $X' \in \operatorname{\mathcal{C}}_{-}$, the profunctor $\mathscr {K}$ induces a homotopy equivalence
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}}( X, X' ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) }( \mathscr {K}(X, -), \mathscr {K}(X',-) ). \]- $(2_ X)$
The vertex $\eta $ is universal.
Proposition 8.3.2.17 will then follow by allowing the triple $(X,Y,\eta )$ to vary.
Condition $(2_ X)$ is the assertion that, for each object $X' \in \operatorname{\mathcal{C}}_{-}$, the composite map
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}}( X, X' ) & \rightarrow & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) }( \mathscr {K}(X, -), \mathscr {K}(X', -) ) \\ & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \mathscr {K}(X,Y), \mathscr {K}(X', Y) ) \\ & \xrightarrow { \circ [\eta ] } & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {K}(X',Y) ) \\ & \simeq & \mathscr {K}(X',Y) \end{eqnarray*}
is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. The equivalence of this assertion with $(1_ X)$ follows immediately from Proposition 8.3.1.3. $\square$
Definition 8.3.2.18 (Balanced Profunctors). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. We say that a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ is balanced if it satisfies the following conditions:
The profunctor $\mathscr {K}$ is representable and corepresentable (Definition 8.3.2.9).
Let $X$ be an object of $\operatorname{\mathcal{C}}_{-}$, let $Y$ be an object of $\operatorname{\mathcal{C}}_{+}$, and let $\eta $ be a vertex of the Kan complex $\mathscr {K}(X,Y)$. Then $\eta $ is universal if and only if it is couniversal.
In other words, $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ is balanced if it satisfies the hypotheses of Proposition 8.3.2.17 both when regarded as a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ and when regarded as a profunctor from $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$ to $\operatorname{\mathcal{C}}^{\operatorname{op}}_{+}$.
Remark 8.3.2.19. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $\lambda $. Then the coupling $\lambda $ is balanced (in the sense of Definition 8.2.6.1) if and only if the profunctor $\mathscr {K}$ is balanced (in the sense of Definition 8.3.2.18). See Remark 8.3.2.8.
Corollary 8.3.2.20. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{C}}_{-}$ to $\operatorname{\mathcal{C}}_{+}$. The following conditions are equivalent:
The profunctor $\mathscr {K}$ is balanced (in the sense of Definition 8.3.2.18).
The $\infty $-category $\operatorname{\mathcal{C}}_{+}$ is locally small and $\mathscr {K}$ induces a fully faithful functor
whose essential image is spanned by the corepresentable functors $\operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$.
The $\infty $-category $\operatorname{\mathcal{C}}_{-}$ is locally small and $\mathscr {K}$ induces a fully faithful functor
whose essential image is spanned by the representable functors $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \rightarrow \operatorname{\mathcal{S}}$.
\[ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, - ), \]
\[ \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad Y \mapsto \mathscr {K}(-,Y), \]
Proof. We will prove the equivalence of $(1)$ and $(2)$; the equivalence of $(1)$ and $(3)$ follows by a similar argument. Assume first that $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ is a balanced profunctor. Invoking Proposition 8.3.2.17, we see that the functor
\[ \Phi : \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, - ) \]
is fully faithful, and that the essential image of $\Phi $ consists of corepresentable functors from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{S}}$. Fix an object $Y \in \operatorname{\mathcal{C}}_{+}$. Since $\mathscr {K}$ is representable, there exists an object $X \in \operatorname{\mathcal{C}}_{-}$ and a universal vertex $\eta \in \mathscr {K}(X,Y)$. Our assumption that $\mathscr {K}$ is balanced guarantees that $\eta $ is also couniversal. In particular, for every object $Y' \in \operatorname{\mathcal{C}}_{+}$, $\eta $ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{+}}(Y,Y') \xrightarrow {\sim } \mathscr {K}(X,Y')$, so that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{+}}(X,Y')$ is essentially small. If $F: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ is any functor corepresented by $Y$, then Theorem 5.6.6.13 guarantees that $F$ is isomorphic to $\mathscr {K}(X, -)$ (as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}})$), and therefore belongs to the essential image of $\Phi $. Allowing the object $Y$ to vary, we deduce that the profunctor $\mathscr {K}$ satisfies condition $(2)$.
We now prove the converse. Assume that the functor $\Phi $ is fully faithful and that the essential image of $\Phi $ is spanned by the corepresentable functors $\operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$. We wish to show that the profunctor $\mathscr {K}$ is balanced. Since $\Phi $ takes values in the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors, the profunctor $\mathscr {K}$ is corepresentable. We next show that $\mathscr {K}$ is representable. Fix an object $Y \in \operatorname{\mathcal{C}}_{+}$; we wish to show that the functor $\mathscr {K}(-, Y): \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable. Since $\operatorname{\mathcal{C}}_{-}$ is locally small, there exists a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by $Y$ (Theorem 5.6.6.13). Then $F$ belongs to the essential image of $\Phi $. We may therefore assume without loss of generality that $F = \mathscr {K}( X_0, -)$ for some object $X_0 \in \operatorname{\mathcal{C}}_{-}$. Choose a couniversal vertex $\eta _0 \in \mathscr {K}(X_0,Y) = F(Y)$. Since $\Phi $ is fully faithful, Proposition 8.3.2.17 implies that $\eta _0$ is also universal, so that $\mathscr {K}(-,Y)$ is representable by $X_0$.
To complete the proof, we must show that the pairing $\mathscr {K}$ satisfies the second condition of Definition 8.3.2.18. Let $Y \in \operatorname{\mathcal{C}}_{+}$ be as above, let $X$ be any object of $\operatorname{\mathcal{C}}_{-}$, and let $\eta $ be a vertex of the Kan complex $\mathscr {K}(X,Y)$. Assume that $\eta $ is universal; we wish to show that it is also couniversal (the reverse implication follows from Proposition 8.3.2.17). Choose $\eta _0 \in \mathscr {K}(X_0, Y)$ as above. Since $\eta _0$ is universal, there exists an isomorphism $u: X \rightarrow X_0$ in the $\infty $-category $\operatorname{\mathcal{C}}_{-}$ such that $\mathscr {K}(u,\operatorname{id}_ Y)( \eta _0 )$ and $\eta $ belong to the same connected component of the Kan complex $\mathscr {K}(X,Y)$ (Remark 5.6.6.6). We may therefore assume without loss of generality that $\eta = \mathscr {K}(u, \operatorname{id}_ Y)(\eta _0)$ (Remark 5.6.6.3). The desired result now follows by applying Remark 5.6.6.4 to the isomorphism of functors $\mathscr {K}(u, -): \mathscr {K}(X_0, -) \rightarrow \mathscr {K}(X, -)$. $\square$
Corollary 8.3.2.21. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category, and let $\operatorname{Fun}^{\mathrm{corep}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors. Then the evaluation map is a balanced profunctor.
\[ \operatorname{ev}: \operatorname{Fun}^{\mathrm{corep}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad (F,C) \mapsto F(C) \]
Remark 8.3.2.22. Up to equivalence, every balanced profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ can be obtained from the construction of Corollary 8.3.2.21. More precisely, let $\operatorname{Fun}^{\mathrm{corep}}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors. If $\mathscr {K}$ is balanced, then it factors as a composition where $\Phi $ is an equivalence of $\infty $-categories by virtue of Corollary 8.3.2.20.
\[ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \xrightarrow {\Phi \times \operatorname{id}} \operatorname{Fun}^{\mathrm{corep}}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) \times \operatorname{\mathcal{C}}_{+} \xrightarrow {\operatorname{ev}} \operatorname{\mathcal{S}}, \]