8.2.6 Balanced Couplings
Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. In this section, we formulate a concrete criterion to determine if $\lambda $ is representable (or corepresentable) by an equivalence of $\infty $-categories.
Definition 8.2.6.1. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. We say that $\lambda $ is balanced if it satisfies the following conditions:
- $(1)$
The coupling $\lambda $ is representable. That is, for each object $C_{+} \in \operatorname{\mathcal{C}}_{+}$, there exists a universal object $C \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{+}(C) = C_{+}$.
- $(2)$
The coupling $\lambda $ is corepresentable. That is, for each object $C_{-} \in \operatorname{\mathcal{C}}_{-}$, there exists a couniversal object $C \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{-}(C) = C_{-}$.
- $(3)$
An object $C \in \operatorname{\mathcal{C}}$ is universal if and only if it is couniversal.
Example 8.2.6.2. For every $\infty $-category $\operatorname{\mathcal{C}}$, the twisted arrow coupling $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ of Example 8.2.0.2 is balanced. See Example 8.2.1.5.
Example 8.2.6.3. Let $X$ be a Kan complex, let $\lambda _{-}: \operatorname{Fun}( \Delta ^1, X) \rightarrow X$ be the morphism given by evaluation at the vertex $0 \in \Delta ^1$, and let $\lambda _{+}: \operatorname{Fun}( \Delta ^1, X) \rightarrow X$ be the morphism given by evaluation at the vertex $1 \in \Delta ^1$. It follows from Corollary 3.1.3.3 that the map
\[ \lambda = (\lambda _{-}, \lambda _{+}): \operatorname{Fun}( \Delta ^1, X) \rightarrow X \times X \]
is a Kan fibration; in particular, we can view it as a coupling of $X$ with itself. For each vertex $x \in X$, the path spaces $\lambda _{-}^{-1} \{ x\} = \{ x\} \operatorname{\vec{\times }}_{X} X$ and $\lambda _{+}^{-1} \{ x\} = X \operatorname{\vec{\times }}_{X} \{ x\} $ are contractible Kan complexes (Example 3.4.1.14), so that every object of $\operatorname{Fun}( \Delta ^1, X)$ is both universal and couniversal for the coupling $\lambda $. In particular, $\lambda $ is a balanced coupling.
We can now formulate the main result of this section.
Theorem 8.2.6.5. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. The following conditions are equivalent:
- $(1)$
The coupling $\lambda $ is balanced.
- $(2)$
The coupling $\lambda $ is representable by an equivalence of $\infty $-categories $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$.
- $(3)$
The coupling $\lambda $ is corepresentable by an equivalence of $\infty $-categories $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$.
Corollary 8.2.6.6. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. Then $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are equivalent if and only if there exists a balanced coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$.
Corollary 8.2.6.7. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. Then $\lambda $ is balanced if and only if there exists an equivalence of couplings
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{\lambda } \ar [r]^-{F} & \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F_{-}^{\operatorname{op}} \times F_{+} } & \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. } \]
Proof.
Suppose that $\lambda $ is balanced. By virtue of Theorem 8.2.6.5, the coupling $\lambda $ is corepresentable by a functor $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ which is an equivalence of $\infty $-categories. We can therefore choose a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{\lambda } \ar [r]^-{F} & \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F_{-}^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} } \]
which exhibits $\lambda $ as corepresented by $F_{-}$. Since $F_{-}$ is an equivalence of $\infty $-categories, it follows that $F$ is also an equivalence of $\infty $-categories (Proposition 4.5.2.21). The reverse implication is an immediate consequence of Example 8.2.6.2 (and Remark 8.2.6.4).
$\square$
We will deduce Theorem 8.2.6.5 from the following more general result.
Proposition 8.2.6.8. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories which is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$. The following conditions are equivalent:
- $(1)$
Every universal object of $\operatorname{\mathcal{C}}$ is couniversal.
- $(2)$
The functor $G$ is fully faithful.
Proof.
Using Proposition 8.2.4.3, we can choose a commutative diagram
8.52
\begin{equation} \begin{gathered}\label{equation:coupling-fully-faithful} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \ar [r]^-{ \widetilde{G}'} \ar [d] & \operatorname{\mathcal{C}}\ar [r]^-{ \widetilde{G} } \ar [d]^{\lambda } & \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \ar [d] \\ \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ \operatorname{id}\times G } & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{-}, } \end{gathered} \end{equation}
where the left square exhibits $\lambda $ as represented by $G$ in the sense of Definition 8.2.4.1, and the right square exhibits $\lambda $ as represented by $G$ in the sense of Definition 8.2.3.5. Invoking (the dual of) Lemma 8.2.3.6, we see that $(1)$ is equivalent to the following:
- $(1')$
The left square of (8.52) is a categorical pullback diagram.
For each object $C \in \operatorname{\mathcal{C}}_{+}$, Proposition 8.2.3.7 guarantees that the composite functor $\widetilde{G} \circ \widetilde{G}'$ carries $\operatorname{id}_{C}$ to an isomorphism of $\operatorname{\mathcal{C}}_{-}$ (regarded as an object of $\operatorname{Tw}(\operatorname{\mathcal{C}}_{-} )$). It follows from Theorem 8.2.2.11 that $(G, \widetilde{G}' \circ \widetilde{G}, G)$ and $(G, \operatorname{Tw}(G), G)$ are isomorphic when viewed as objects of $\operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}), \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}))$ (since both are initial objects of the $\infty $-category $\operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}), \operatorname{Tw}(\operatorname{\mathcal{C}}_{-})) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-})} \{ G\} $). In particular, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}_{+}$, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{+}}(X,Y) \ar [d]^{G} \ar [r] & \{ X\} \times _{ \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \times _{\operatorname{\mathcal{C}}_{+}} \{ Y\} \ar [d]^{ \widetilde{G}' \circ \widetilde{G} } \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{-}}( G(X), G(Y) ) \ar [r] & \{ G(X) \} \times _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \times _{\operatorname{\mathcal{C}}_{-}} \{ G(Y) \} } \]
commutes up to homotopy, where the horizontal maps are the homotopy equivalences of Notation 8.1.2.14. Using Corollary 5.1.7.16, we see that $(2)$ is equivalent to the following:
- $(2')$
The outer rectangle of (8.52) is a categorical pullback diagram.
The equivalence of $(1')$ and $(2')$ is a special case of Proposition 4.5.2.18, since the right square of (8.52) is a categorical pullback square by assumption.
$\square$
Proof of Theorem 8.2.6.5.
We will prove the equivalence $(1) \Leftrightarrow (2)$; the equivalence $(1) \Leftrightarrow (3)$ follows by a similar argument. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is a coupling of $\infty $-categories which is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$. Combining Theorem 8.2.5.1 with Proposition 8.2.6.8, we see that $\lambda $ is balanced if and only if the following conditions are satisfied:
The functor $G$ is fully faithful.
The functor $G$ admits a left adjoint $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$.
The functor $F$ is fully faithful.
It follows from Corollary 6.2.2.21 that these conditions are satisfied if and only if $G$ is an equivalence of $\infty $-categories.
$\square$