Recall that if $V$ is a finite-dimensional vector space and $f \colon V \to V$ is an endomorphism, we can define $$ \tr(f \vert V) \in \End(1). $$ For $A$ a $k$-algebra and $F$ a bi-module, we also could define $$ HH(A, F) = A \otimes_{A \otimes A^\mathrm{rev}} F. $$
Categorically, for $C \in \mathsf{LinCat}_ \Lambda$ that is dualizable and $F \colon C \to C$, this was $$ \tr(F \vert C) \in \End_{\mathsf{LinCat}_ \Lambda} 1 = \mathsf{Mod}_ \Lambda. $$ So for $A \in \mathsf{Alg}(\mathsf{LinCat}_ \Lambda)$ and $F$ a $A$-bimodule, we can get $$ HH(A, F) = A \otimes_{A \otimes A^\mathrm{rev}} F \in \mathsf{LinCat}_ \Lambda. $$
The relative tensor product#
Let $R$ be a symmetric monoidal category with geometric realizations and tensor products commute with geometric realizations. Let $A \in \mathsf{Alg}(R)$. Then we get a functor $$ \mathsf{LMod}_ {A^\mathrm{rev}}(R) \times \mathsf{LMod}_ A(R) \to R; \quad (N, M) \mapsto N \otimes_R M. $$ This can be characterized by the universal property, but it can also be concretely constructed using the bar construction, $$ N \otimes_A M = \varprojlim N \otimes A^{\otimes n} \otimes M. $$
We can enhance this when these have actions on the other side. We similarly have $$ {}_ C \mathsf{Mod}_ A \times {}_ A \mathsf{Mod}_ B \to {}_ C \mathsf{Mod}_ B. $$
Example 1. If $A = \mathsf{LMod}_ R$ for some $\mathbb{E}_ 2$-algebra $A$, and $M$ is some $A$-bimodule, then $$ HH(A, F) \in \mathsf{LinCat}_ \Lambda $$ will be $\mathsf{LMod}_ {HH(R, M)}$.
Informally, here is what we are doing. We have a fully faithful embedding $$ \mathsf{Morita}(\mathsf{Vect}) \to \mathsf{LinCat}_ \Lambda. $$ There is supposed to be also a fully faithful embedding $$ \mathsf{Morita}(\mathsf{LinCat}_ \Lambda) \to 2\mathsf{-LinCat}_ \Lambda. $$ This $2\mathsf{-LinCat}_ \Lambda$ will be some kind of $(\infty, 3)$-category where objects are $(\infty, 2)$-categories with all the hom being a $\Lambda$-linear category, satisfying some nice properties.
Some functoriality#
Let $A, B$ be two algebras and let ${}_ A M_B$ be an $A$-$B$-bimodule.
Definition 2. We say that $M$ is left-dualizable if there exists $N$ a $B$-$A$-bimodule together with unit and evalaution maps $$ u_M \colon B \to N \otimes_A M, \quad e_M \colon M \otimes_B N \to A $$ satisfying the usual Zorro relations.
Remark 3. An $A$-$B$-bimodule $M$ is left dualizable if and only if $$ M \otimes_B (-) \colon \mathsf{LinCat}_ B \to \mathsf{LinCat}_ A $$ admits a right adjoint.
Now consider
- $(A, F_A)$ and $(B, F_B)$,
- $M$ a left-dualizable $A$-$B$-bimodule,
- $\eta \colon M \otimes_B F_B \to F_A \otimes_A M$.
From this data, we get a map $$ \begin{align} HH(B, F_B) &= B \otimes_{B \otimes B^\mathrm{rev}} F_B \to (N \otimes_A M) \otimes_{A \otimes A^\mathrm{rev}} F_B = A \otimes_{A \otimes A^\mathrm{rev}} (M \otimes_B F_B \otimes_B N) \br &\to A \otimes_{A \otimes A^\mathrm{rev}} (F_A \otimes_A M \otimes_B N) \to A \otimes_{A \otimes A^\mathrm{rev}} (F_A \otimes_A A) = HH(A, F_A). \end{align} $$
Example 4. Let $R = \mathsf{LinCat}_ \Lambda$. If $A = F_A = \mathsf{Mod}_ \Lambda$, and $B = F_B = \mathsf{Mod}_ \Lambda$, then $M$ is left-dualizable as an $A$-$B$-bimodule if and only if $M$ is dualizable in $\mathsf{Mod}_ \Lambda$. In this case, $$ \tr(M, \eta) \colon HH(\mathsf{Mod}_ \Lambda) = \mathsf{Mod}_ \Lambda \to \mathsf{Mod}_ \Lambda = HH(\mathsf{Mod}_ \Lambda) $$ is just the usual $HH(M, \Lambda)$.
Definition 5. Let $A \in \mathsf{Alg}(R)$ for $R$ some symmetric monoidal category. We say that $A$ is smooth when $$ A \in {}_ {A \otimes A^\mathrm{rev}} \mathsf{BMod}_ {1_R} $$ is left-dualizable. We say that $A$ is proper when $$ A \in {}_ {1_R} \mathsf{BMod}_ {A^\mathrm{rev} \otimes A} $$ is left-dualizable. We say that it is 2-dualizable when it is both smooth and proper.
When it proper, we have some $S_A$ a $A^\mathrm{rev} \otimes A$-$1_R$-bimodule together with $$ A \otimes A^\mathrm{rev} \to S_A \otimes A, \quad A \otimes_{A \otimes A^\mathrm{rev}} S_A \to 1_R. $$ This $S_A$ is called the Serre module.
For $A$ that is 2-dualizable, there is a trace formula again.
Theorem 6. Assume $A$ is 2-dualizable, and let $F_1, F_2$ be $A$-$A$-bimodules that are left-dualizable. Given a $$ \alpha \colon F_1 \otimes_A F_2 \cong F_2 \otimes_A F_1 $$ we get an equivalence $$ \tr(HH(F_2, \alpha^{-1}) \vert HH(A, F_1)) \simeq \tr(HH(F_1, \alpha) \vert HH(A, F_2)). $$
Going to 3-categories#
Unfortunately, I don’t have good interesting examples of 3-dualizable objects. All of them seem to have some finiteness conditions, such as fusion categories.
Definition 7. Let $M$ be a left dualizable $A$-$B$-bimodule in $\mathsf{LinCat}_ \Lambda$. We say $M$ is
- left-smooth if $u_M \colon B \to N \otimes_A M$ admits a $B$-bilinear right adjoint,
- left-proper if $e_M$ admits a $A$-linear right adjoint.
Example 8. Let $A = B = \mathsf{Mod}_ \Lambda$. This recovers the previous definition of smooth proper $\Lambda$-linear categories.
Example 9. If $B = \mathsf{Mod}_ \Lambda$ and $M = A$ as a left $A$-module, then $M$ is always dualizable as an $A$-$1$-bimodule with $N = A$, with $$ u_M \colon \mathsf{Mod}_ \Lambda \to A \otimes_A A = A $$ being the unit and $$ e_M \colon A \otimes_A A \to A $$ being the multiplication. Then $M = A$ is left-proper if $e_M$ admits a continuous right adjoint, and left-smooth if $1_A$ is compact.
Definition 10. We say $A$ a $\Lambda$-linear monoidal category is $semi-rigid* if multiplication $A \otimes A \to A$ admits a continuous right adjoint. We say that it is (locally) rigid if it is semi-rigid and $1_A$ is compact.
Proposition 11. If $A \in \mathsf{Alg}(\mathsf{LinCat}_ \Lambda)$ is compactly generated, then $A$ is (locally) rigid if and only if every compact object admits both left and right duals.
So $\mathsf{QCoh}(X)$ for $X$ quasi-compact quasi-separated are examples. Another example is $A = \mathsf{Ind}(D_\mathrm{cons}(B \backslash G / B))$.
Coming back to $(A, F_A)$ and $(B, F_B)$, let $M$ be a left-proper and left-smooth $A$-$B$-bimodule, with $$ \eta \colon M \otimes_B F_B \to F_A \otimes_A M $$ that admits a continuous right adjoint.
Proposition 12. The induced $$ HH(M, \eta) \colon HH(B, F_B) \to HH(A, F_A) $$ now admits a continuous right adjoint.
Corollary 13. If $(B, F_B) = (\mathsf{Mod}_ \Lambda, \mathsf{Mod}_ \Lambda)$, then $M$ is just a left $A$-module and $\eta \colon M \to F_A \otimes_A M$. Then we get $$ HH(M, \eta) \colon \mathsf{Mod}_ \Lambda \to HH(A, F_A) $$ which corresponds to some object $[M, \eta]_ {F_A}$, which is like the Chern character.
Now if $M$ is left-smooth and left-proper, then $\eta^R$ is continuous, hence $[M, \eta]_ {F_A}$ is compact in $HH(A, F_A)$.
Example 14. If $A$ is rigid and taking $M = A$, we see that $\eta$ corresponds to $X \in F_A$. Then $[M, \eta]_ {F_A}$ is the image of $X$ under the functor $$ \tr \colon F_A \to HH(A, F_A). $$ If $X$ is compact, then $\tr(X)$ is compact.