Monoidal categories

• Examples of traces
• Monoidal categories

Examples of traces#

Let’s recall this category $\mathsf{Morita}(\mathsf{Vect}_ k^\heartsuit)$ where every object is dualizable. For any associative $k$-algebra $A$, we can take $A^\mathrm{rev}$ as the dual, with the bimodules $$ {}_ {A \otimes A^\mathrm{rev}} A_k, \quad {}_ k A_{A^\mathrm{rev} \otimes A}. $$ For any endomorphism of $A$, which is a bimodule ${}_ A M_A$, we can consider $$ \tr({}_ A M_A \vert A) = A \otimes_{A \otimes A^\mathrm{rev}} (M \otimes A^\mathrm{rev}) \otimes_{A \otimes A^\mathrm{rev}} A \in \mathsf{Vect}_ k^\heartsuit. $$ This can be thought of as “tensoring over a circle.” Then this trace can be identified as $$ A \otimes_{A \otimes A^\mathrm{rev}} M = M / \operatorname{span}\lbrace am - ma \rbrace. $$ Instead of $\mathsf{Vect}_ k^\heartsuit$ we can use $\mathsf{Vect}_ k$ and then we obtain Hochschild homology.

Similarly, we had the category of correspondences $\mathsf{Corr}(\mathsf{Sch}_ {/S})$. Here, every $X$ is again dualizable with dual $X$. The counit and unit maps be $$ X \times_S X \xleftarrow{\Delta} X \to S, \quad S \leftarrow X \xrightarrow{\Delta} X \times_S X. $$ Given any $f \colon X \to X$, we can consider the endomorphism $$ X \xleftarrow{f} X = X, $$ and the trace will be $$ \tr(f \vert X) = X^f \in \mathsf{Sch}_ {/S} = \End(S). $$

Let’s look at the cohomological correspondences $\mathsf{CohCorr}(\mathsf{Sch}_ {/S}^\mathrm{ft})$ where $S/k$ is of finite type and coefficients are in $\mathbb{F}_ l$ where $l$ is invertible in $k$. The objects are pairs $(X, \mathscr{F})$ where $X \in \mathsf{Sch}_ {/S}$ and $\mathscr{F} \in D(\mathsf{Shv}(X_\mathrm{et}, \mathbb{F}_ l)^\heartsuit)$. Here, not every object is dualizable.

Theorem 1. An object $(X, \mathscr{F})$ is dualizable if and only if $\mathscr{F}$ is universally locally acyclic with respect to $X \to S$.

Now let’s consider the category $\mathsf{LinCat}_ k$. A category of the form $$ \mathcal{C} = D(\mathsf{QCoh}(X)^\heartsuit) $$ for $X$ a quasi-compact quasi-separated scheme, is dualizable with the dual being $\mathcal{C}$ itself. One functor will be $$ \mathcal{C} \otimes \mathcal{C} \to \mathsf{Vect}_ k; \quad \mathscr{F} \boxtimes \mathscr{G} \mapsto R\Gamma(X, \mathscr{F} \otimes \mathscr{G}), $$ and the other functor will involve Grothendieck duality.

Any endomorphism $f \colon X \to X$ of schemes defines a functor $$ Lf^\ast \colon D(\mathsf{QCoh}(X)^\heartsuit) \to D(\mathsf{QCoh}(X)^\heartsuit). $$ What is the trace of this functor? We will see that $$ \tr(Lf^\ast \vert D(\mathsf{QCoh}(X)^\heartsuit)) = R\Gamma(X^f, \mathscr{O}) \in \mathsf{Vect}_ k, $$ where $X^f$ is to be understood in terms of derived geometry.

Later we will see that any sheaf theory gives a symmetric monoidal functor from $\mathsf{Corr}(\mathsf{Sch}_ {/S}) \to \mathsf{LinCat}_ k$. Then we can compute these things by functoriality of trace.

Monoidal categories#

Definition 2. A semigroup (non-unital monoidal) category is a triple $(\mathcal{C}, \otimes ,a)$ where

• $\mathcal{C}$ is a category,
• $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a functor,
• the diagram $$ \begin{CD} \mathcal{C} \times \mathcal{C} \times \mathcal{C} @>{\otimes \times \id}>> \mathcal{C} \times \mathcal{C} \br @V{\id \times \otimes}VV @V{\otimes}VV \br \mathcal{C} \times \mathcal{C} @>{\otimes}>> \mathcal{C} \end{CD} $$ has $a$ as a natural equivalence between them, i.e., there is a functorial $a_{X,Y,Z} \colon (X \otimes Y) \otimes Z \cong X \otimes (Y \otimes Z)$,
• satisfying the property that the pentagonal diagram $$ \begin{CD} ((X \otimes Y) \otimes Z) \otimes W @>{\cong}>> (X \otimes Y) \otimes (Z \otimes W) @>{\cong}>> X \otimes (Y \otimes (Z \otimes W)) \br @VVV @. @AAA \br (X \otimes (Y \otimes Z)) \otimes W @>{\cong}>> \longrightarrow @>{\cong}>> X \otimes ((Y \otimes Z) \otimes W) \end{CD} $$ commutes.

Definition 3. A unit in a non-unital monoidal category $\mathcal{C}$ is a pair $(1, u)$ where

• $1 \in \mathcal{C}$ is an object,
• $u \colon 1 \otimes 1 \cong 1$ is an isomorphism,

such that both functors $$ 1 \otimes - \colon \mathcal{C} \to \mathcal{C}, \quad - \otimes 1 \colon \mathcal{C} \to \mathcal{C} $$ are fully faithful.

Proposition 4. Let $(1, u)$ be a unit in $(\mathcal{C}, \otimes, a)$.

1. There are natural equivalences $$ l \colon 1 \otimes - \simeq \id_\mathcal{C}, \quad r \colon - \otimes 1 \simeq \id_\mathcal{C}. $$
2. The following diagrams are commutative. $$ \begin{CD} (X \otimes 1) \otimes Y @>{\cong}>> X \otimes (1 \otimes Y) \br @V{r_X \otimes \id_Y}VV @V{\id_X \otimes l_Y}VV \br X \otimes Y @= X \otimes Y \end{CD} $$ $$ \begin{CD} (X \otimes Y) \otimes 1 @>{\cong}>> X \otimes (Y \otimes 1) \br @V{r_{X \otimes Y}}VV @V{\id_X \otimes r_Y}VV \br X \otimes Y @= X \otimes Y \end{CD} $$ $$ \begin{CD} (1 \otimes X) \otimes Y @>{\cong}>> 1 \otimes (X \otimes Y) \br @V{l_X \otimes \id_Y}VV @V{l_{X \otimes Y}}VV \br X \otimes Y @= X \otimes Y \end{CD} $$
3. $l_1 = r_1 = u$.
4. The monoid $\End_\mathcal{C}(1)$ is a commutative monoid.
5. There are no nontrivial automorphisms of $(1, u)$.
6. If $(1^\prime, u^\prime)$ is another unit, then there exists a unique $\epsilon \colon (1, u) \cong (1^\prime, u^\prime)$.

Definition 5. A monoidal category is a non-unital monoidal category that has a unit.