# Kerodon

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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:

$(1)$

The profunctor $\mathscr {K}$ is balanced (in the sense of Definition 8.3.2.18).

$(2)$

The $\infty$-category $\operatorname{\mathcal{C}}_{+}$ is locally small and $\mathscr {K}$ induces a fully faithful functor

$\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, - ),$

whose essential image is spanned by the corepresentable functors $\operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$.

$(3)$

The $\infty$-category $\operatorname{\mathcal{C}}_{-}$ is locally small and $\mathscr {K}$ induces a fully faithful functor

$\operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad Y \mapsto \mathscr {K}(-,Y),$

whose essential image is spanned by the representable functors $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \rightarrow \operatorname{\mathcal{S}}$.

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 5.7.6.13 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 5.7.6.13). 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 5.7.6.6). We may therefore assume without loss of generality that $\eta = \mathscr {K}(u, \operatorname{id}_ Y)(\eta _0)$ (Remark 5.7.6.3). The desired result now follows by applying Remark 5.7.6.4 to the isomorphism of functors $\mathscr {K}(u, -): \mathscr {K}(X_0, -) \rightarrow \mathscr {K}(X, -)$. $\square$