Symmetric monoidal categories

Let’s start with some examples of monoidal categories.

All previous examples are monoidal categories, in particular, symmetric monoidal categories.
Let $M$ be a commutative monoid. From this we can define a category $BM$ with one object and $\End(\ast) = M$. We really need $M$ to be commutative, because we saw last time that the endomorphisms of $1$ all commute with each other. This is even a symmetric monoidal category.
Let $\mathcal{C}$ be a category with finite products. Then $\mathcal{C}$ admits a natural symmetric monoidal structure given by $X \otimes Y = X \times Y$ and the unit $1$ being the final object.
For $\mathcal{C}$ a category, we can define the category $\End(\mathcal{C})$ of functors from $\mathcal{C}$ to itself. This is naturally a monoidal category, with $\otimes$ being composition. This is not symmetric monoidal. One can define a notion of a module category over a monoidal category, and $\mathcal{C}$ will naturally be a module over $\End(\mathcal{C})$.
Let $\Gamma$ be a finite group. We can look at $\mathcal{C} = \mathsf{Vect}^\Gamma$ the category of $\Gamma$-graded vector spaces. We can give it a monoidal structure with $$ (V \ast W)_ {\gamma^{\prime\prime}} = \bigoplus_{\gamma \gamma^\prime = \gamma^{\prime\prime}} V_\gamma \otimes W_{\gamma^\prime}. $$ Here is a fancier way of thinking about this. We can think of $\mathsf{Vect}^\Gamma$ as a sheaf of $k$-vector spaces on the discrete space $\Gamma$. If $m \colon \Gamma \times \Gamma \to \Gamma$ is the multiplication map, this is $$ \mathscr{V} \ast \mathscr{W} = m_\ast(\mathscr{V} \boxtimes \mathscr{W}). $$
We can generalize the previous example as follows. Suppose $\mathcal{C}$ is a category of “spaces,” and $\mathcal{C}$ has finite limits. Let $\mathcal{D}$ be a “sheaf theory” on $\mathcal{C}$. For $f \colon X \to Y$ a map in $\mathcal{C}$, we can consider $Z = X \times_Y X$. We can form the correspondence $$ Z \leftarrow{p = \id \times f \times \id} X \times_Y X \times_Y X \xrightarrow{q = \id \times \Delta_X \times \id} (X \times_Y X) \times (X \times_Y X) = Z \times Z. $$ Then we can define a monoidal structure on $\mathcal{D}(Z)$ by $$ \mathscr{F} \ast \mathscr{G} = p_! q^\ast (\mathscr{F} \boxtimes \mathscr{G}). $$
Another example of the previous setup is the Hecke category. For $G$ a finite group and $H \subseteq G$ a finite subgroup, we can take $X = BH \to Y = BG$. Then $Z = H \backslash G / H$. It is a good exercise to try and write down $\ast$ explicitly.

Duals in a monoidal category#

Let $\mathcal{C}$ be a monoidal category.

Definition 1. Let $X \in \mathcal{C}$ be an object. A right dual of $X$ consists of $(X^\vee, u_X, e_X)$, where
$X^\vee \in \mathcal{C}$ is an object,
$u_X \colon 1 \to X \otimes X^\vee$ is a morphism,
$e_X \colon X^\vee \otimes X \to 1$ is a morphism,
such that $$ X \cong 1 \otimes X \to X \otimes X^\vee \otimes X \to X \otimes 1 \cong X $$ and $$ X^\vee \cong X^\vee \otimes 1 \to X^\vee \otimes X \otimes X^\vee \to 1 \otimes X^\vee \cong X^\vee $$ are both identity morphisms.
A left dual of $X$ is a triple $({}^\vee X, {}^\vee u_X, {}^\vee e_X)$ such that $(X, {}^\vee u_X, {}^\vee e_X)$ is a right dual of $X^\vee$.

In general, $X^\vee$ and ${}^\vee X$ need not be isomorphic, assuming they both exist. Also, having a dual is much weaker than being invertible, because we don’t impose the condition that $u_X \colon 1 \to X \otimes X^\vee$ be an isomorphism.

Proposition 2. Let $(X^\vee, u_X, e_X)$ be a right dual of $X$. Then there exist canonical isomorphisms $$ \Hom(Y \otimes X, Z) \cong \Hom(Y, Z \otimes X^\vee) $$ and $$ \Hom(Y, X \otimes Z) \cong \Hom(X^\vee \otimes Y, Z). $$

Proof.

There are natural maps $$ \Hom(Y \otimes X, Z) \xrightarrow{- \otimes X^\vee} \Hom(Y \otimes X \otimes X^\vee, Z \otimes X^\vee) \xrightarrow{u_X} \Hom(Y, Z \otimes X^\vee) $$ and also $$ \Hom(Y, Z \otimes X^\vee) \xrightarrow{- \otimes X} \Hom(Y \otimes X, Z \otimes X^\vee \otimes X) \xrightarrow{e_X} \Hom(Y \otimes X, Z). $$ We claim that these two maps are inverses to each other. Given any $f \colon Y \otimes X \to Z$, it is sent to $$ Y \otimes X \xrightarrow{\id \otimes u_X \otimes \id} Y \otimes X \otimes X^\vee \otimes X \xrightarrow{f \otimes \id \otimes \id} Z \otimes X^\vee \otimes X \xrightarrow{\id \otimes e_X} Z. $$ This is the same as $f$, because we can replace the composition of the last two maps with $$ Y \otimes X \otimes X^\vee \otimes X \xrightarrow{\id \otimes \id \otimes e_X} Y \otimes X \xrightarrow{f} Z, $$ and then $u_X \otimes \id$ composed with $\id \otimes e_X$ is just the identity.

Another way of saying this is that $- \otimes X$ admits a right adjoint $- \otimes X^\vee$ and $X \otimes -$ admits a left adjoint $X^\vee \otimes -$.

Corollary 3. If a right (respectively, left) dual exists, then it is unique up to unique isomorphism.

Corollary 4. Assume that $X$ has a right dual. Then $- \otimes X$ preserves all colimits, and $X \otimes -$ preserves all limits.

Assume that $X$ and $Y$ both have right duals. Then for any $f \colon X \to Y$, we can define a dual $$ f^\vee \colon Y^\vee \to Y^\vee \otimes X \otimes X^\vee \to Y^\vee \otimes Y \otimes X^\vee \to X^\vee. $$

Corollary 5. If every object in $\mathcal{C}$ admits a right dual, then the functor $$ \mathcal{C} \to \mathcal{C}^\mathrm{op} \colon X \mapsto X^\vee $$ is fully faithful.

Corollary 6. Let $X, Y \in \mathcal{C}$ be objects with right duals $X^\vee, Y^\vee$. Then the following are equivalent:
$g = f^\vee$,
the following diagram is commutative, $$ \begin{CD} Y^\vee \otimes X @>{1 \otimes f}>> Y^\vee \otimes Y \br @V{g \otimes 1}VV @V{e_Y}VV \br X^\vee \otimes X @>{e_X}>> 1 \end{CD} $$
the following diagram is commutative. $$ \begin{CD} 1 @>{u_Y}>> Y \otimes Y^\vee \br @V{u_X}VV @V{1 \otimes g}VV \br X \otimes X^\vee @>{f \otimes 1}>> Y \otimes X^\vee \end{CD} $$

Definition 7. A monoidal category $\mathcal{C}$ is said to be rigid if every object $X \in \mathcal{C}$ admits both a left dual and right dual.

Corollary 8. In a rigid monoidal category $\mathcal{C}$, the functors $- \otimes X$ and $X \otimes -$ preserves all limits and colimits. Moreover, $\sigma_C \colon \mathcal{C} \to \mathcal{C}$ sending $X$ to $(X^\vee)^\vee$ is an equivalence.

Symmetric monoidal categories#

To define a notion of trace, we need to switch the order of $X$ and $X^\vee$. So we need a bit more than just a dual.

Definition 9. A symmetric monoidal category consists of $(\mathcal{C}, \otimes, a, c)$ where $(\mathcal{C}, \otimes, a)$ is a monoidal category (implicit in here is a unit), and $c \colon X \otimes Y \cong Y \otimes X$ is a functorial isomorphism in $X, Y$ such that
$X \otimes Y \xrightarrow{c_{X,Y}} Y \otimes X \xrightarrow{c_{Y,X}} X \otimes Y$ is the identity,
the diagram $$ \begin{CD} (X \otimes Y) \otimes Z @>{a}>> X \otimes (Y \otimes Z) @>{c}>> X \otimes (Z \otimes Y) \br @V{c}VV @. @V{c}VV \br (Y \otimes X) \otimes Z @>{c}>> Z \otimes (Y \otimes X) @>{a}>> (Z \otimes Y) \otimes X \end{CD} $$ commutes.

In a symmetric monoidal category, a right dual is the same as a left dual. So we can talk about duals without distinguishing between them.

Definition 10. Let $f \colon X \to X$ be an endomorphism of a dualizable object in a symmetric monoidal category $\mathcal{C}$. Then we define $$ \tr(f \vert X) = 1 \xrightarrow{u_X} X \otimes X^\vee \xrightarrow{f \otimes 1} X \otimes X^\vee \xrightarrow{c} X^\vee \otimes X \xrightarrow{e_X} 1. $$

Proposition 11. Let $X, Y \in \mathcal{C}$ be dualizable objects in a symmetric monoidal category $\mathcal{C}$.
$\tr(f \vert X) = \tr(f^\vee \otimes X)$ for $f \colon X \to X$.
$\tr(g \circ f \vert X) = \tr(f \circ g \vert Y)$ for $f \colon X \to Y$ and $g \colon Y \to X$.
$\tr(f \otimes g \vert X \otimes Y) = \tr(f \vert X) \tr(g \vert Y)$ for $f \colon X \to X$ and $g \colon Y \to Y$.
If $\mathcal{C}$ is additive, then $\tr(f \oplus g \vert X \oplus Y) = \tr(f \vert X) + \tr(g \vert Y)$.

Internal hom in a symmetric monoidal category#

Definition 12. Let $X, Y \in \mathcal{C}$ be objects in $\mathcal{C}$ a symmetric monoidal category. An internal hom object $\underline{\Hom}(X, Y)$ is an object $\mathcal{C}$ representing the functor $$ Z \mapsto \Hom_\mathcal{C}(Z \otimes X, Y), $$ i.e., we have $$ \Hom(Z, \underline{\Hom}(X, Y)) = \Hom(Z \otimes X, Y). $$ We write $X^\ast = \underline{\Hom}(X, 1)$ if it exists.

Lemma 13. Let $\mathcal{C}$ be a symmetric monoidal category.
If $X \in \mathcal{C}$ is dualizable, then $\underline{\Hom}(X, Y)$ exists and is isomorphic to $X^\vee \otimes Y$.
An object $X \in \mathcal{C}$ is dualizable if $X^\ast$ and $\underline{\Hom}(X, X)$ exists, and moreover the natural map $X^\ast \otimes X \to \underline{\Hom}(X, X)$ is an isomorphism.

Here, the natural map $X^\ast \otimes X \to \underline{\Hom}(X, X)$ comes from the following construction. First there is always an evaluation map $$ \mathrm{ev}_ X \colon X^\ast \otimes X \to 1 $$ coming from the identity map on $X^\ast$, and then we can use the composition $$ X^\ast \otimes X \otimes X \xrightarrow{\id \otimes c} X^\ast \otimes X \otimes X \xrightarrow{\mathrm{ev}_ X \otimes \id} X $$ and use tensor-hom adjunction to get the map.

Definition 14. A symmetric monoidal category $\mathcal{C}$ is said to be closed if the internal hom $\underline{\Hom}(X, Y)$ exists for all $X, Y \in \mathcal{C}$.

Examples include $\mathsf{Vect}_ k^\heartsuit$, $D(\mathsf{Vect}_ k^\heartsuit)$, and $\mathsf{LinCat}_ k$.