Definition 8.2.0.1. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. A coupling of $\operatorname{\mathcal{C}}_{+}$ with $\operatorname{\mathcal{C}}_{-}$ is an $\infty $-category $\operatorname{\mathcal{C}}$ equipped with a left fibration $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$.
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8.2 Couplings of $\infty $-Categories
We now axiomatize an essential feature of the twisted arrow construction introduced in §8.1.
In the situation of Definition 8.2.0.1, we will often refer to the functor $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ as a coupling of $\infty $-categories. This terminology signifies both that $\lambda $ is a left fibration and that its target is equipped with a specified factorization as a product of $\infty $-categories $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$.
Example 8.2.0.2 (The Twisted Arrow Coupling). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the map $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ of Notation 8.1.1.6 is a coupling of $\operatorname{\mathcal{C}}$ with itself (Proposition 8.1.1.11). We will refer to $\lambda $ as the twisted arrow coupling of the $\infty $-category $\operatorname{\mathcal{C}}$.
Construction 8.2.0.3. Let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor of $\infty $-categories. Pulling back the left fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}_{-} ) \rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{-}$ of Proposition 8.1.1.11, we obtain a left fibration of $\infty $-categories which we regard as a coupling of $\operatorname{\mathcal{C}}_{+}$ with $\operatorname{\mathcal{C}}_{-}$. We will refer to $\lambda _{G}$ as the coupling associated to the functor $G$.
\[ \lambda _{G}: \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \times _{ \operatorname{\mathcal{C}}_{-} } \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}, \]
We say that a coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is representable if, for every object $C_{+} \in \operatorname{\mathcal{C}}_{+}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ C_{+} \} $ has an initial object (Definition 8.2.1.3). It is not difficult to show that, for every functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$, the coupling $\lambda _{G}$ of Construction 8.2.0.3 is representable (Variant 8.2.1.6). Our primary goal in this section is to prove the converse:
Theorem 8.2.0.4. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. Then the assignment $G \mapsto \lambda _{G}$ of Construction 8.2.0.3 induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$} \} / \textnormal{Isomorphism} \ar [d] \\ \{ \textnormal{Representable couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$} \} / \textnormal{Equivalence}. } \]
Remark 8.2.0.5. Let $\operatorname{\mathcal{C}}_{-}$ be a (locally small) $\infty $-category. For every $\infty $-category $\operatorname{\mathcal{C}}_{+}$, Corollary 5.6.0.6 supplies an identification of equivalence classes of couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ (having essentially small fibers) with isomorphism classes of functors $T: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. Moreover, $\lambda $ is representable if and only if $T$ factors through the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ spanned by the representable functors (see Proposition 5.6.6.21). Consequently, Theorem 8.2.0.4 supplies a bijection It is not hard to see that this bijection depends functorially on $\operatorname{\mathcal{C}}_{+}$, and is therefore induced by an isomorphism $\operatorname{\mathcal{C}}_{-} \simeq \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ in the homotopy category $\mathrm{h} \mathit{ \operatorname{QCat}}$. We can therefore regard Theorem 8.2.0.4 as an “implicit” version of Yoneda's lemma. We will give a more precise formulation in §8.3 (see Theorem 8.3.3.13).
\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} ) \xrightarrow {\sim } \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{C}}_{+}, \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}}) ). \]
Let us outline our approach to Theorem 8.2.0.4. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. For a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$, we say that $\lambda $ is representable by $G$ if it is equivalent to the coupling $\lambda _{G}$ of Construction 8.2.0.3 (Definition 8.2.3.1). Theorem 8.2.0.4 asserts that every representable coupling is representable by some functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$, which is uniquely determined up to isomorphism. To prove this, we need to construct a commutative diagram
8.28
\begin{equation} \begin{gathered}\label{equation:exhibits-as-represented-ur} \xymatrix@R =50pt@C=50pt{ \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]^-{ \operatorname{id}\times G } & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{-} } \end{gathered} \end{equation}
which is a categorical pullback square; in this case, we say that (8.28) exhibits $\lambda $ as represented by $G$ (Definition 8.2.3.5).
It will be useful to place this problem in a somewhat larger context. Suppose that $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ is another coupling of $\infty $-categories. We will refer to a commutative diagram
8.29
\begin{equation} \begin{gathered}\label{equation:exhibits-as-represented-ur2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{\lambda } & \operatorname{\mathcal{D}}\ar [d]^{\mu } \\ \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}}_{+} } \end{gathered} \end{equation}
as a morphism of couplings from $\lambda $ to $\mu $ (Definition 8.2.2.1). The collection of such diagrams can be organized into an $\infty $-category $\operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, which is equipped with a forgetful functor
\[ \Phi : \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-})^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ). \]
It is not difficult to see that $\Phi $ is also a left fibration: that is, it can be regarded as a coupling of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} )$ with the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )$ (Proposition 8.2.2.2). Suppose now that the coupling $\lambda $ is representable, and that the coupling $\mu $ is corepresentable (that is, for every object $D_{-} \in \operatorname{\mathcal{D}}_{-}$, the $\infty $-category $\{ D_{-} \} \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}$ has an initial object). In §8.2.2, we show these these assumptions imply that the coupling $\Phi $ is also corepresentable (Theorem 8.2.2.11). In particular, every functor $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{D}}_{-}$ has a canonical promotion to a commutative diagram of the form (8.29), which is characterized (up to isomorphism) by the requirement that it represents an initial object of the $\infty $-category $\{ F_{-} \} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. In §8.2.3, we specialize this assertion to the situation where $\mu $ is the twisted arrow coupling $\operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ and $F_{-}$ is the identity functor from $\operatorname{\mathcal{C}}_{-}$ to itself. In this case, we obtain a diagram of the form (8.28) and use it to deduce Theorem 8.2.0.4.
Every assertion in the preceding discussion has a dual counterpart, where the assumption of representability is replaced by corepresentability (and vice versa). If $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is a corepresentable coupling of $\infty $-categories, then Theorem 8.2.0.4 guarantees the existence of a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ \widetilde{F} } \ar [d]^{\lambda } & \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}. } \]
for some functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$, which is uniquely determined up to isomorphism; in this case, we say that the coupling $\lambda $ is corepresentable by $F$ (Variant 8.2.3.8). In §8.2.5, we study couplings which are simultaneously representable and corepresentable. Our main result asserts that if a coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$, then it is corepresentable by a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ if and only if $F$ is left adjoint to $G$ (Theorem 8.2.5.1). Our proof is based on an alternative characterization of the corepresenting functor $F$, which we explain in §8.2.4.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the twisted arrow coupling $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ has the following features:
- $(a)$
The coupling $\lambda $ is corepresentable. That is, for every object $X \in \operatorname{\mathcal{C}}$, the $\infty $-category $\{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}})$ has an initial object.
- $(b)$
The coupling $\lambda $ is representable. That is, for every object $Y \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} $ has an initial object.
- $(c)$
Let $f$ be an object of the $\infty $-category $\operatorname{Tw}(\operatorname{\mathcal{C}})$, which we regard as a morphism $X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $f$ is initial when viewed as an object of the $\infty $-category $\{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}})$ if and only if it is initial when viewed as an object of the $\infty $-category $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} $ (by virtue of Corollary 8.1.2.21, both conditions are equivalent to the requirement that $f$ corresponds to an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$).
In §8.2.6, we show that twisted arrow couplings are characterized (up to equivalence) by these properties. More precisely, we show that a coupling $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ satisfies conditions $(a)$, $(b)$ and $(c)$ if and only if it is representable (or corepresentable) by an equivalence of $\infty $-categories (Theorem 8.2.6.5), and therefore equivalent to the twisted arrow $\infty $-category associated to the $\infty $-category $\operatorname{\mathcal{D}}_{-}$ (or $\operatorname{\mathcal{D}}_{+}$). In this case, we say that the coupling $\lambda $ is balanced (Definition 8.2.6.1).