Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

8.2.3 Representations of Couplings

We now apply Theorem 8.2.2.9 to give a classification of representable couplings.

Definition 8.2.3.1. Let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor of $\infty $-categories. We will say that a coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is representable by $G$ if it is equivalent (as a left fibration over $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$) to the coupling $\lambda _{G}$ of Construction 8.2.0.3.

Remark 8.2.3.2. Let $G,G': \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be functors which are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} )$). Then a coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is representable by $G$ if and only if it is representable by $G'$. See Proposition 5.1.6.5.

Example 8.2.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the twisted arrow coupling $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ of of Example 8.2.0.2 is representable by the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.

Our goal is to prove the following restatement of Theorem 8.2.0.4:

Theorem 8.2.3.4. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. Then $\lambda $ is representable (in the sense of Definition 8.2.1.3) if and only there exists a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ such that $\lambda $ is representable by $G$ (in the sense of Definition 8.2.3.1). If this condition is satisfied, then the functor $G$ is uniquely determined up to isomorphism.

Before giving the proof of Theorem 8.2.3.4, it will be useful to formulate a more precise version of Definition 8.2.3.1.

Definition 8.2.3.5. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor. We say that a morphism of couplings

8.21
\begin{equation} \begin{gathered}\label{equation:duality-functor-right-to-left} \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}

exhibits the coupling $\lambda $ as represented by $G$ if it is a categorical pullback square. Note that $\lambda $ is representable by $G$ if and only if there exists a morphism of couplings which exhibits $\lambda $ as represented by $G$.

We now describe an alternative formulation of Definition 8.2.3.5.

Lemma 8.2.3.6. Suppose we are given a morphism of couplings

8.22
\begin{equation} \begin{gathered}\label{equation:duality-functor-left-to-right-preliminary} \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}

where $\lambda $ is representable and $F_{-}$ is an equivalence of $\infty $-categories. The following conditions are equivalent:

$(1)$

The diagram (8.22) is a categorical pullback square.

$(2)$

For every object $Y \in \operatorname{\mathcal{C}}_{+}$, the functor $F$ induces an equivalence of $\infty $-categories

\[ F_{Y}: \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ Y \} \rightarrow \operatorname{\mathcal{D}}\times _{ \operatorname{\mathcal{D}}_{+} } \{ F_{+}(Y) \} . \]
$(3)$

For every universal object $C \in \operatorname{\mathcal{C}}$, the image $F(C) \in \operatorname{\mathcal{D}}$ is universal.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Theorem 5.1.5.1 and Remark 8.2.2.4. To complete the proof, it will suffice to show that for each object $Y \in \operatorname{\mathcal{C}}_{+}$, the following conditions are equivalent:

$(2_ Y)$

The functor $F_{Y}$ is an equivalence of $\infty $-categories.

$(3_ Y)$

The functor $F_{Y}$ carries initial objects of $\operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ Y \} $ to initial objects of $\operatorname{\mathcal{D}}\times _{ \operatorname{\mathcal{D}}_{+} } \{ F_{+}(Y) \} $.

Replacing $\operatorname{\mathcal{D}}$ by the $\infty $-category $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times _{\operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}$, we can assume that $F_{-}$ is an isomorphism. In this case, the equivalence of $(2_ Y)$ and $(3_ Y)$ is a special case of Corollary 4.6.6.22. $\square$

Proposition 8.2.3.7. Suppose we are given a morphism of couplings

8.23
\begin{equation} \begin{gathered}\label{equation:duality-functor-left-to-right2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ \widetilde{G} } \ar [d]^{\lambda } & \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \ar [d]^{\mu } \\ \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}

The following conditions are equivalent:

$(1)$

The diagram (8.23) exhibits the coupling $\lambda $ as represented by the functor $G$ (in the sense of Definition 8.2.3.1).

$(2)$

For every object $C \in \operatorname{\mathcal{C}}_{+}$, the functor $\widetilde{G}$ induces an equivalence of $\infty $-categories

\[ \widetilde{G}_{C}: \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ C \} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \times _{ \operatorname{\mathcal{C}}_{-} } \{ G(C) \} . \]
$(3)$

The coupling $\lambda $ is representable and, for every universal object $C \in \operatorname{\mathcal{C}}$, the image $\widetilde{G}(C) \in \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} )$ is an isomorphism (when viewed as a morphism of the $\infty $-category $\operatorname{\mathcal{C}}_{-}$).

$(4)$

The coupling $\lambda $ is representable and the triple $(\operatorname{id}, \widetilde{G}, G)$ is initial when viewed as an object of the $\infty $-category $\{ \operatorname{id}\} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{-} )^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} ) )$.

Proof. The implication $(1) \Rightarrow (2)$ is immediate. Note that, if condition $(2)$ is satisfied, then the coupling $\lambda $ is representable; the implications $(2) \Rightarrow (3) \Rightarrow (1)$ then follow from Lemma 8.2.3.6 (using the characterization of universal objects of $\operatorname{Tw}(\operatorname{\mathcal{C}}_{-} )$ given by Example 8.2.1.5). The equivalence $(3) \Leftrightarrow (4)$ follows from Theorem 8.2.2.9. $\square$

Proof of Theorem 8.2.3.4. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. It follows from Proposition 8.2.3.7 that $\lambda $ is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ if and only if it is representable and $G$ can be lifted to an initial object of the $\infty $-category $\operatorname{\mathcal{E}}= \{ \operatorname{id}\} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{-} )^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} ) )$. This immediately shows that $G$ is uniquely determined up to isomorphism. To prove existence, it suffices to show that if $\lambda $ is representable then $\operatorname{\mathcal{E}}$ has an initial object. This follows from Theorem 8.2.2.9. $\square$

Variant 8.2.3.8. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ be a functor. We say that $\lambda $ is corepresentable by $F$ if there exists a categorical pullback square

8.24
\begin{equation} \begin{gathered}\label{equation:duality-functor-left-to-right} \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}}_{+}. } \end{gathered} \end{equation}

In this case, we will say that the diagram (8.24) exhibits the coupling $\lambda $ as corepresented by $F$. It follows from Theorem 8.2.3.4 that $\lambda $ is corepresentable (in the sense of Definition 8.2.1.3) if and only if it is corepresentable by $F$, for some functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$. Moreover, if this condition is satisfied, then the functor $F$ is uniquely determined up to isomorphism.

Couplings representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ can be characterized by a universal mapping property.

Proposition 8.2.3.9. Let $\mu = (\mu _{-}, \mu _{+}): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ be a coupling of $\infty $-categories, let $G: \operatorname{\mathcal{D}}_{+} \rightarrow \operatorname{\mathcal{D}}_{-}$ be a functor, and suppose we are given a morphism of couplings

8.25
\begin{equation} \begin{gathered}\label{equation:exhibits-as-representable-version1} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}\ar [r]^-{ \widetilde{G} } \ar [d] & \operatorname{Tw}( \operatorname{\mathcal{D}}_{-} ) \ar [d] \\ \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+} \ar [r]^-{ \operatorname{id}\times G } & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{-}. } \end{gathered} \end{equation}

The following conditions are equivalent:

$(1)$

The diagram (8.25) exhibits $\mu $ as represented by $G$ (in the sense of Definition 8.2.3.5).

$(2)$

For every coupling of $\infty $-categories $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ and every pair of functors $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{D}}_{-}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{D}}_{+}$, composition with $\widetilde{G}$ induces a homotopy equivalence of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \{ F_{-} \} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+}) } \{ F_{+} \} \ar [d] \\ \{ F_{-} \} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{D}}_{-} )) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{-}) } \{ G \circ F_{+} \} } \]
$(3)$

For every coupling of $\infty $-categories $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ and every functor $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{D}}_{+}$, composition $\widetilde{G}$ induces an equivalence of $\infty $-categories

\[ \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+}) } \{ F_{+} \} \rightarrow \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{D}}_{-} )) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{-}) } \{ G \circ F_{+} \} . \]

Proof. We first show that $(1) \Rightarrow (2)$. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{D}}_{-}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{D}}_{+}$ be functors. If condition $(1)$ is satisfied, then Remark 4.5.2.8 guarantees that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [r]^-{\widetilde{G} \circ } \ar [d]^{\mu \circ } & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{D}}_{-} ) ) \ar [d] \\ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} ) \times \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} ) \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} ) \times \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{-} ) } \]

is a categorical pullback square, where the vertical maps are left fibrations (Corollary 4.2.5.2). Assertion $(2)$ now follows by applying Corollary 4.5.2.25 to the object $( F_{-}^{\operatorname{op}} \circ \lambda _{-}, F_{+} \circ \lambda _{+} ) \in \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} ) \times \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} )$. The implication $(2) \Rightarrow (3)$ follows by applying Corollary 5.1.5.4 to the commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=-20pt{ \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+}) } \{ F_{+} \} \ar [rr] \ar [dr] & & \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{D}}_{-} )) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{-}) } \{ G \circ F_{+} \} \ar [dl] \\ & \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}}, & } \]

since the vertical maps are left fibrations (Proposition 8.2.2.2). We complete the proof by showing that $(3)$ implies $(1)$. Specializing assertion $(3)$ to the coupling $\lambda : \Delta ^0 \xrightarrow {\sim } (\Delta ^0)^{\operatorname{op}} \times \Delta ^0$, we deduce that $\widetilde{G}$ induces an equivalence of $\infty $-categories $\operatorname{\mathcal{D}}\times _{ \operatorname{\mathcal{D}}_{+} } \{ D\} \rightarrow \operatorname{Tw}( \operatorname{\mathcal{D}}_{-} ) \times _{ \operatorname{\mathcal{D}}_{-} } \{ G(D) \} $ for each object $D \in \operatorname{\mathcal{D}}_{+}$, so that (8.25) is a categorical pullback square by virtue of Proposition 8.2.3.7. $\square$