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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.

Definition 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 $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$.

In the situation of Definition, 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 (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 is a coupling of $\operatorname{\mathcal{C}}$ with itself (Proposition We will refer to $\lambda $ as the twisted arrow coupling of the $\infty $-category $\operatorname{\mathcal{C}}$.

Construction 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, we obtain a left fibration of $\infty $-categories

\[ \lambda _{G}: \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \times _{ \operatorname{\mathcal{C}}_{-} } \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}, \]

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 $G$.

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 It is not difficult to show that, for every functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$, the coupling $\lambda _{G}$ of Construction is representable (Variant Our primary goal in this section is to prove the converse:

Theorem Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. Then the assignment $G \mapsto \lambda _{G}$ of Construction 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 Let $\operatorname{\mathcal{C}}_{-}$ be a (locally small) $\infty $-category. For every $\infty $-category $\operatorname{\mathcal{C}}_{+}$, Corollary 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 Consequently, Theorem supplies a bijection

\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} ) \xrightarrow {\sim } \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{C}}_{+}, \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}}) ). \]

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 as an “implicit” version of Yoneda's lemma. We will give a more precise formulation in §8.3 (see Theorem

Let us outline our approach to Theorem 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 (Definition Theorem 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

\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

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

\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 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 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 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

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 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 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 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:


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.


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.


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, 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, 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


  • Subsection 8.2.1: Representable Couplings
  • Subsection 8.2.2: Morphisms of Couplings
  • Subsection 8.2.3: Representations of Couplings
  • Subsection 8.2.4: Presentations of Representable Couplings
  • Subsection 8.2.5: Adjunctions as Couplings
  • Subsection 8.2.6: Balanced Couplings