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6.1.6 Duality in Monoidal Categories

We now specialize the theory of adjunctions to the setting of $2$-categories of the form $B\operatorname{\mathcal{C}}$ (Example 2.2.2.4), where $\operatorname{\mathcal{C}}$ is a monoidal category. Throughout this section, we write ${\bf 1}$ for the unit object of a monoidal category $\operatorname{\mathcal{C}}$.

Definition 6.1.6.1. Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$. A duality datum is a pair $(\operatorname{coev}, \operatorname{ev})$, where $\operatorname{coev}: {\bf 1} \rightarrow Y \otimes X$ and $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ are morphisms of $\operatorname{\mathcal{C}}$ satisfying the following compatibility conditions:

$(Z1)$

The composition

\[ X \xrightarrow [\sim ]{\rho _{X}^{-1}} X \otimes {\bf 1} \xrightarrow {\operatorname{id}_ X \otimes \operatorname{coev}} X \otimes (Y \otimes X) \xrightarrow [\sim ]{\alpha _{X,Y,X}} (X \otimes Y) \otimes X \xrightarrow {\operatorname{ev}\otimes \operatorname{id}_ X} {\bf 1} \otimes X \xrightarrow [\sim ]{\lambda _ X} X \]

is equal to the identity morphism $\operatorname{id}_{X}$. Here the isomorphism $\alpha _{X,Y,X}$ is the associativity constraint for the monoidal category $\operatorname{\mathcal{C}}$, and the isomorphisms $\lambda _{X}$ and $\rho _{X}$ are the left and right unit constraints of Construction 2.1.2.17.

$(Z2)$

The composition

\[ Y \xrightarrow [\sim ]{\lambda _{Y}^{-1} } {\bf 1} \otimes Y \xrightarrow {\operatorname{coev}\otimes \operatorname{id}_ Y} (Y \otimes X) \otimes Y \xrightarrow [\sim ]{\alpha _{Y,X,Y}^{-1}} Y \otimes (X \otimes Y) \xrightarrow {\operatorname{id}_{Y} \otimes \operatorname{ev}} Y \otimes {\bf 1} \xrightarrow [\sim ]{\rho _ Y} Y \]

is equal to the identity morphism $\operatorname{id}_{Y}$.

If these conditions are satisfied, then we will refer to $\operatorname{coev}$ as the coevaluation morphism of the duality datum $(\operatorname{coev}, \operatorname{ev})$, and to $\operatorname{ev}$ as the evaluation morphism of the duality datum $(\operatorname{coev}, \operatorname{ev})$. In this case, we say that the pair $(\operatorname{coev}, \operatorname{ev})$ exhibits $X$ as a left dual of $Y$, also also that it exhibits $Y$ as a right dual of $X$.

Remark 6.1.6.2 (Duals as Adjoints). Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$, which we regard as $1$-morphisms of the $2$-category $B\operatorname{\mathcal{C}}$ described in Example 2.2.2.4. Suppose we are given a pair of morphisms

\[ \operatorname{coev}: {\bf 1} \rightarrow Y \otimes X \quad \quad \operatorname{ev}: X \otimes Y \rightarrow {\bf 1} \]

in $\operatorname{\mathcal{C}}$, which we identify with $2$-morphisms of $B\operatorname{\mathcal{C}}$. Then the pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum in the monoidal category $\operatorname{\mathcal{C}}$ (in the sense of Definition 6.1.6.1) if and only if it is an adjunction in the $2$-category $B\operatorname{\mathcal{C}}$ (in the sense of Definition 6.1.1.1).

Remark 6.1.6.3 (Adjoints as Duals). Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, let $f,g: X \rightarrow X$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{X} \rightarrow g \circ f$ and $\epsilon : f \circ g \rightarrow \operatorname{id}_{X}$ be $1$-morphisms of $\operatorname{\mathcal{C}}$. Then the pair $(\eta , \epsilon )$ is an adjunction in the $2$-category $\operatorname{\mathcal{C}}$ (in the sense of Definition 6.1.1.1) if and only if it is a duality datum in the monoidal category $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ of Remark 2.2.1.7.

Remark 6.1.6.4. Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$ and morphisms

\[ \operatorname{coev}: {\bf 1} \rightarrow Y \otimes X \quad \quad \operatorname{ev}: X \otimes Y \rightarrow {\bf 1}. \]

Then:

  • The pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum in the monoidal category $\operatorname{\mathcal{C}}$ if and only if it is a duality datum in the reverse monoidal category $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ of Example 2.1.3.5. Note that passage to the reverse monoidal category reverses the roles of $X$ and $Y$: if $X$ is the left dual of $Y$ in the monoidal category $\operatorname{\mathcal{C}}$, then it is the right dual of $Y$ in the monoidal category $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ (and vice-versa).

  • The pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum in $\operatorname{\mathcal{C}}$ if and only if the pair $(\operatorname{ev}^{\operatorname{op}}, \operatorname{coev}^{\operatorname{op}})$ is a duality datum in the opposite monoidal category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (see Example 2.1.3.4). Note that passage to the opposite monoidal category reverses the roles of evaluation and coevaluation: $\operatorname{ev}^{\operatorname{op}}$ is the coevaluation morphism for the duality datum $(\operatorname{ev}^{\operatorname{op}}, \operatorname{coev}^{\operatorname{op}})$, while $\operatorname{coev}^{\operatorname{op}}$ is the evaluation morphism. Similarly, if $X$ is the left dual of $Y$ in the monoidal category $\operatorname{\mathcal{C}}$, then it is the right dual of $Y$ in the opposite monoidal category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (and vice-versa).

Proposition 6.1.6.5. Let $\operatorname{\mathcal{C}}$ be a monoidal category and let $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

  • For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the composite map

    \begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, Y \otimes D) & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X \otimes C, X \otimes (Y \otimes D) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X \otimes C, (X \otimes Y) \otimes D) \\ & \xrightarrow {\operatorname{ev}} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X \otimes C, {\bf 1} \otimes D) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X \otimes C, D ) \end{eqnarray*}

    is a bijection.

  • For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the composite map

    \begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, D \otimes X) & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \otimes Y, (D \otimes X) \otimes Y ) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \otimes Y, D \otimes (X \otimes Y)) \\ & \xrightarrow {\operatorname{ev}} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \otimes Y, D \otimes {\bf 1} ) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C \otimes Y, D ) \end{eqnarray*}

    is a bijection.

  • There exists a morphism $\operatorname{coev}: {\bf 1} \rightarrow Y \otimes X$ for which the pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum, in the sense of Definition 6.1.6.1.

Moreover, if these conditions are satisfied, then the morphism $\operatorname{coev}: {\bf 1} \rightarrow Y \otimes X$ is unique.

Proof. Apply Proposition 6.1.2.13 to the $2$-category $B\operatorname{\mathcal{C}}$ of Example 2.2.2.4. $\square$

Definition 6.1.6.6. Let $\operatorname{\mathcal{C}}$ be a monoidal category. We will say that a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ in $\operatorname{\mathcal{C}}$ is a duality datum if it satisfies the equivalent conditions of Proposition 6.1.6.5: that is, if there exists a morphism $\operatorname{coev}: {\bf 1} \rightarrow Y \otimes X$ for which the pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum in the sense of Definition 6.1.6.1.

Applying Proposition 6.1.6.5 to the opposite monoidal category $\operatorname{\mathcal{C}}^{\operatorname{op}}$, we obtain the following:

Variant 6.1.6.7. Let $\operatorname{\mathcal{C}}$ be a monoidal category and let $\operatorname{coev}: {\bf 1} \rightarrow Y \otimes X$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

  • For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the composite map

    \begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X \otimes C, D) & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y \otimes (X \otimes C), Y \otimes D) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (Y \otimes X) \otimes C, Y \otimes D)) \\ & \xrightarrow {\operatorname{coev}} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( {\bf 1} \otimes C, Y \otimes D) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, Y \otimes D ) \end{eqnarray*}

    is a bijection.

  • For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the composite map

    \begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \otimes Y, D) & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (C \otimes Y) \otimes X, D \otimes X ) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \otimes (Y \otimes X), D \otimes X) \\ & \xrightarrow {\operatorname{coev}} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \otimes {\bf 1}, D \otimes X ) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, D \otimes X ) \end{eqnarray*}

    is a bijection.

  • There exists a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ for which the pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum, in the sense of Definition 6.1.6.1.

Moreover, if these conditions are satisfied, then the morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ is unique.

Definition 6.1.6.8. Let $\operatorname{\mathcal{C}}$ be a monoidal category. We will say that a morphism $\operatorname{coev}: {\bf 1} \rightarrow Y \otimes X$ in $\operatorname{\mathcal{C}}$ is a duality datum if it satisfies the equivalent conditions of Variant 6.1.6.7: that is, if there exists a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ for which the pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum in the sense of Definition 6.1.6.1.

Definition 6.1.6.9. Let $\operatorname{\mathcal{C}}$ be a monoidal category. Then:

  • We say that an object $X \in \operatorname{\mathcal{C}}$ is right dualizable if there exists an object $Y \in \operatorname{\mathcal{C}}$ and a duality datum $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$. In this case, we will also say that $Y$ is a right dual of $X$, or that the morphism $\operatorname{ev}$ exhibits $Y$ as a right dual of $X$.

  • We say that an object $Y \in \operatorname{\mathcal{C}}$ is left dualizable if there exists an object $X \in \operatorname{\mathcal{C}}$ and a duality datum $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$. In this case, we will also say that $X$ is a left dual of $Y$, or that the morphism $\operatorname{ev}$ exhibits $X$ as a left dual of $Y$.

Example 6.1.6.10. Let $\operatorname{\mathcal{C}}$ be a monoidal category. We say that an object $X \in \operatorname{\mathcal{C}}$ is invertible if there exists an object $Y \in \operatorname{\mathcal{C}}$ such that the tensor products $Y \otimes X$ and $X \otimes Y$ are isomorphic to the unit object ${\bf 1}$. If this condition is satisfied, then any choice of isomorphism ${\bf 1} \simeq Y \otimes X$ is a duality datum (this is a special case of Proposition 6.1.4.1). In particular, the object $Y$ is a right dual of $X$. Similarly, $Y$ is a left dual of $X$.

Exercise 6.1.6.11. Let $\operatorname{\mathcal{C}}$ be a category which admits finite products, and regard $\operatorname{\mathcal{C}}$ as equipped with the monoidal structure given by cartesian products (Example 2.1.3.2). Show that an object $X \in \operatorname{\mathcal{C}}$ is left (or right) dualizable if and only if it is isomorphic to the final object ${\bf 1}$.

Exercise 6.1.6.12. Let $k$ be a field and let $\operatorname{Vect}_{k}$ denote the category of vector spaces over $k$, equipped with the monoidal structure described in Example 2.1.3.1. Show that an object $V \in \operatorname{Vect}_{k}$ is left (or right) dualizable if and only if it is finite-dimensional as a vector space over $k$.

It is instructive to contrast Definition 6.1.6.6 with a slightly more general notion of duality.

Definition 6.1.6.13. Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$. We will say that a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ exhibits $Y$ as a weak right dual of $X$ if, for every object $W \in \operatorname{\mathcal{C}}$, the composite map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X \otimes W, X \otimes Y) \xrightarrow {\operatorname{ev}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X \otimes W, {\bf 1} ) \]

is bijective. We say that $\operatorname{ev}$ exhibits $X$ as a weak left dual of $Y$ if, for every object $Z \in \operatorname{\mathcal{C}}$, the composite map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Z,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Z \otimes Y, X \otimes Y) \xrightarrow {\operatorname{ev}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Z \otimes Y, {\bf 1} ) \]

is bijective.

Remark 6.1.6.14. Let $\operatorname{\mathcal{C}}$ be a monoidal category and let $X$ be an object of $\operatorname{\mathcal{C}}$. It follows immediately from the definition that if there exists a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ which exhibits $Y$ as a weak right dual of $X$, then the pair $(Y, \operatorname{ev})$ is unique up to isomorphism and depends functorially on $X$. To emphasize this dependence we will sometimes denote the object $Y$ by $X^{\vee }$ and abuse terminology by referring to it as the weak right dual of $X$.

Similarly, if $Y$ is a fixed object of $\operatorname{\mathcal{C}}$ and there exists a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ which exhibits $X$ as a weak left dual of $Y$, then the pair $(X,\operatorname{ev})$ is uniquely determined up to isomorphism and depends functorially on $Y$. We will emphasize this dependence by denoting the object $X$ by ${^{\vee }Y}$ and referring to it as the weak left dual of $Y$.

Proposition 6.1.6.15. Let $\operatorname{\mathcal{C}}$ be a monoidal category and let $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ be a morphism of $\operatorname{\mathcal{C}}$. Then:

$(1)$

If the morphism $\operatorname{ev}$ exhibit $Y$ as a right dual of $X$ (Definition 6.1.6.6), then it exhibits $Y$ as a weak right dual of $X$ (Definition 6.1.6.13). The converse holds if $X$ is right dualizable.

$(2)$

If the morphism $\operatorname{ev}$ exhibits $X$ as a left dual of $Y$, then it exhibits $X$ as a weak left dual of $Y$. The converse holds if $Y$ is left dualizable.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. If $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ is a duality datum, then it exhibits $Y$ as a weak right dual of $X$ by virtue of Variant 6.1.3.2 (applied to the $2$-category $B\operatorname{\mathcal{C}}$). Conversely, suppose that $\operatorname{ev}$ exhibits $Y$ as a weak right dual of $X$. If there exists another object $Y' \in \operatorname{\mathcal{C}}$ and a duality datum $\operatorname{ev}': X \otimes Y' \rightarrow {\bf 1}$, then the universal property of $Y$ guarantees that there is a unique morphism $u: Y' \rightarrow X$ for which $\operatorname{ev}'$ is equal to the composite map $X \otimes Y' \xrightarrow {\operatorname{id}_ X \otimes u} X \otimes Y \xrightarrow {\operatorname{ev}} {\bf 1}$. Since $\operatorname{ev}'$ exhibits $Y'$ as a weak right dual of $X$, the morphism $u$ must be an isomorphism, so that the morphism $\operatorname{ev}$ is also a duality datum. $\square$

Warning 6.1.6.16. In the situation of Proposition 6.1.6.15, it is possible for an object $X \in \operatorname{\mathcal{C}}$ to admit a weak right dual which is not a right dual. For example, let $\operatorname{\mathcal{C}}= \operatorname{Vect}_{k}$ be the category of vector spaces over a field $k$, equipped with the monoidal structure of Example 2.1.3.1. Let $V$ be a vector space over $k$ and let $V^{\ast } = \operatorname{Hom}_{k}(V, k)$ be its dual space. Then the evaluation map

\[ \operatorname{ev}: V \otimes _{k} V^{\ast } \rightarrow k \quad \quad v \otimes \lambda \mapsto \lambda (v) \]

exhibits $V^{\ast }$ as a weak (right) dual of $V$ (in the sense of Definition 6.1.6.13). However, it is a duality datum only when $V$ is finite-dimensional over $k$ (Exercise 6.1.6.12).

Remark 6.1.6.17. Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$. If both $X$ and $Y$ are right dualizable, then the tensor product $X \otimes Y$ is also right dualizable; moreover we have a canonical isomorphism $(X \otimes Y)^{\vee } \simeq Y^{\vee } \otimes X^{\vee }$ (see Corollary 6.1.5.4 for a more precise statement). Similarly, if both $X$ and $Y$ are left dualizable, then the tensor product $X \otimes Y$ is left dualizable, and there is a canonical isomorphism ${^{\vee }(X \otimes Y)} \simeq {^{\vee }Y} \otimes {^{\vee }X}$.

Exercise 6.1.6.18. Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$. Show that, if $X$ is weakly right dualizable and $Y$ is right dualizable, then the tensor product $X \otimes Y$ is weakly right dualizable (and that there is a canonical isomorphism $(X \otimes Y)^{\vee } \simeq Y^{\vee } \otimes X^{\vee }$).