Let $V$ be a finite-dimensional vector space. Recall that we can define a trace of a map $f \colon V \to V$ by the composition $$ k \to V \otimes V^\vee \xrightarrow{f \otimes \id} V \otimes V^\vee \to k. $$ The upshot is that $\tr(f \vert V)$ can be defined in an arbitrary category $\mathcal{C}$ equipped with $\otimes$, $1$, etc., assuming that $$ 1 \to X \otimes X^\vee, \quad X^\vee \otimes X \to 1 $$ are defined.
Symmetric monoidal categories#
Definition 1 (informal). A symmetric monoidal category is a tuple $(\mathcal{C}, \otimes, a, c, 1, u)$ of
- $\mathcal{C}$ a category,
- $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ a functor,
- a functorial isomorphism $a_{X,Y,Z} \colon (X \otimes Y) \otimes Z \cong X \otimes (Y \otimes Z)$ in $X, Y, Z \in \mathcal{C}$,
- a functorial isomorphism $c_{X,Y} \colon X \otimes Y \cong Y \otimes X$ in $X, Y \in \mathcal{C}$,
- $1 \in \mathcal{C}$ an object,
- an isomorphism $u \colon 1 \otimes 1 \cong 1$,
satisfying a bunch of properties, such as $1 \otimes - \colon \mathcal{C} \to \mathcal{C}$ is an equivalence.
It will automatically follow that there is a functorial isomorphism $1 \otimes X \cong X$, because $$ 1 \otimes (1 \otimes X) \cong (1 \otimes 1) \otimes X \cong 1 \otimes X. $$
Example 2. Here are some examples of symmetric monoidal categories.
- There is the category $\mathsf{Vect}_ k^\heartsuit$ of $k$-vector spaces, with the usual tensor product over $k$.
- We can also look at the category $(\mathsf{Vect}_ k^\mathrm{gr})^\heartsuit$ of graded vector spaces. Here, the tensor product is defined as $$ (V \otimes W)^i = \bigoplus_j V^{i-j} \otimes W^j, $$ and for commutativity we will put in a sign $$ V^i \otimes W^j \cong W^j \otimes V^i; \quad v \otimes w \mapsto (-1)^{ij} w \otimes w. $$
- There is also look at the category of chain complexes, or the derived category.
- In all the above examples, we can replace $k$ by any commutative ring.
- For $X$ a topological space (or even a topos), we can take sheaves of $A$-modules, $\mathcal{C} = \mathsf{Shv}(X, A)^\heartsuit$.
- For $X$ a (quasi-compact quasi-separated) scheme, we can look at $\mathsf{QCoh}(X)^\heartsuit$ the category of quasi-coherent sheaves on $X$.
Example 3. Here is some less familiar example $\mathsf{Morita}(\mathrm{Vect}_ k^\heartsuit)$. The objects are associative $k$-algebras, a morphism from $A$ to $B$ is a $A$-$B$-bimodule. Compositions of morphisms $M$ a $A$-$B$-bimodule and $N$ a $A$-$C$-bimodule is given by $$ M \otimes_B N $$ which is an $A$-$C$-bimodule. This has a symmetric monoidal structure with tensor product given by $(A, B) \mapsto A \otimes_k B$.
In fact, instead of $\mathsf{Vect}_ k^\heartsuit$, we can feed in any symmetric monoidal category $\mathcal{C}$.
Example 4. There is the category $\mathsf{Corr}(\mathsf{Sch}_ k)$ of correspondences. The objects are quasi-compact quasi-separated schemes over $k$, a morphism from $Y$ to $X$ is given by a diagram $$ X \leftarrow C \to Y. $$ Composition here is given by taking the fiber product $$ \begin{CD} X @<<< C @<<< C \times_Y D \br @. @VVV @VVV \br @. Y @<<< D \br @. @. @VVV \br @. @. Z. \end{CD} $$ The tensor product is defined as $(X, Y) \mapsto X \times_k Y$.
Example 5. We can upgrade the previous example to $\mathsf{CohCorr}(\mathsf{Sch}_ S^\mathrm{ft})$. The objects are $(X, \mathscr{F})$ where $X$ is a scheme of finite type over $S$ and $\mathscr{F}$ is an étale sheaf on $X$ with $\mathbb{F}_ l$-coefficients.
A morphism from $(Y, \mathscr{G})$ to $(X, \mathscr{F})$ is a correspondence $$ X \xleftarrow{c_\leftarrow} C \xrightarrow{c_\rightarrow} Y $$ together with a map $$ (c_\leftarrow)_ ! (c_\rightarrow)^\ast \mathscr{G} \to \mathscr{F}. $$ The tensor product is just given by $$ ((X, \mathscr{F}), (Y, \mathscr{G})) \mapsto (X \times Y, \mathscr{F} \boxtimes \mathscr{G}). $$
Example 6. The final example, which will be the focus of the earlier parts of the course, is the category $\mathsf{LinCat}_ k$ of $k$-linear presentable stable ∞-categories. (Examples include $D(\mathsf{Vect}_ k^\heartsuit)$ or $D(\mathsf{QCoh}(X)^\heartsuit)$.) The morphisms are given by $k$-linear colimit-preserving functors. The tensor product here is called the Deligne–Lurie tensor product. For associative algebras $A$ and $B$, this will send $(\mathsf{Mod}_ A, \mathsf{Mod}_ B)$ to $\mathsf{Mod}_ {A \otimes_k B}$.
Dualizable objects#
Let $(\mathcal{C}, \otimes, a, c, 1, u)$ be a symmetric monoidal category.
Definition 7. A dual of $X$ is a triple $(X^\vee, u_X, e_X)$ where
- $X^\vee \in \mathcal{C}$ is an object,
- $u_X \colon 1 \to X \otimes X^\vee$ is a morphism,
- $e_X \colon X \otimes X^\vee \to 1$ is a morphism,
such that $$ X \cong 1 \otimes X \to (X \otimes X^\vee) \otimes X \cong X \otimes (X^\vee \otimes X) \to X \otimes 1 \cong X $$ and a similar map $$ X^\vee \to X^\vee \otimes X \otimes X^\vee \to X^\vee $$ is the identity map.
If a dual exists, it is unique up to unique isomorphism.
Definition 8. An object $X$ is called dualizable if it has a dual.
Definition 9. Let $X$ be a dualizable object in $\mathcal{C}$, and let $f \colon X \to X$ be an endomorphism. Then we define its trace as $$ \tr(f \vert X) = (1 \to X \otimes X^\vee \xrightarrow{f \otimes \id} X \otimes X^\vee \cong X^\vee \otimes X \to 1) \in \End_\mathcal{C}(1). $$
Example 10. Let’s work out the dualizable objects in the above list of examples.
- Inside $\mathcal{C} = \mathsf{Vect}_ k^\heartsuit$, an object $V$ is dualizable if and only if it is finite-dimensional. Then the trace is the usual trace.
- Inside $\mathcal{C} = (\mathsf{Vect}_ k^\mathrm{gr})^\heartsuit$, an object is dualizable if the total dimension is finite, and the trace is the alternating sum $$ \tr(f \vert V^\bullet) = \sum_{i}^{} (-1)^i \tr(f^i \vert V^i). $$ The sign comes from having to swap the order of $X$ and $X^\vee$ in the definition of the trace.
- In the derived category $D(\mathsf{Vect}_ k^\heartsuit)$, an object $V \in \mathcal{C}$ is dualizable if and only if $V$ is perfect. The trace is then a similar alternating sum. A similar thing holds for $D(\mathsf{Mod}_ A^\heartsuit)$ or $D(\mathsf{QCoh}(X)^\heartsuit)$.
Example 11. In the Morita category $\mathsf{Morita}(D(\mathsf{Vect}_ k^\heartsuit))$, every object turns out to be dualizable, with dual being the opposite algebra $A^\mathrm{rev}$. The unit map is the $k$-$A \otimes A^\mathrm{rev}$-bimodule $A$, and the counit map is $A^\mathrm{rev} \otimes A$-$k$-bimodule $A$. Then for any $A$-$A$-bimodule $M$, the trace is given by Hochschild homology $$ \tr(f_M \vert A) = A \otimes_{A \otimes A^\mathrm{rev}}^\mathbb{L} M = HH_\bullet(A, M). $$