Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that:
- $\mathcal C$ is closed under coequalizers. (I generally assume moreover that all of my categories are closed under all colimits.)
- the tensor structure distributes over coequalizers. (I generally assume moreover that my monoidal categories are closed monoidal, meaning that $X\otimes$ has a left adjoint for all objects $X$; this implies that tensoring distributes over all colimits.)
Suppose that $X \in \mathcal C$ is a dualizable object, with dual $X^*$. Then the endomorphism algebra of $X$ is $A = X^* \otimes X \in \mathcal C$, with multiplication $$ A \otimes A = X^* \otimes X \otimes X^* \otimes X \overset{ X^* \otimes \mathrm{ev} \otimes X}\longrightarrow X^* \otimes \mathbf 1 \otimes X = A $$ and unit $\mathrm{coev}: \mathbf 1 \to X^* \otimes X = A$. Note in particular that $A$ is a unital associative algebra object in $\mathcal C$.
For any associative algebra $B \in \mathcal C$, recall that a left module for $B$ is an object $M \in \mathcal C$ along with an action $B \otimes M \to M$ which is compatible with the unit and multiplication on $B$ in the natural way. Similarly, one can define bimodules. Suppose that $M$ is a $B$–$B'$ bimodule, and $M'$ is a $B'$–$B''$ bimodule. The bimodule tensor product is $$ M \underset{B'}\otimes M' = \mathrm{coequalizer}\bigl( M \otimes B' \otimes M' \rightrightarrows M \otimes M' \bigr) $$ where the two arrows are the actions of $B'$ either on $M$ or $M'$. The result is a $B$–$B''$ bimodule. My requests above assure that this coequalizer exists, and moreover that bimodule tensor product is associative.
This allows one to define the Morita bicategory of $\mathcal C$, whose objects are associative algebras in $\mathcal C$, 1-morphisms are bimodules, and 2-morphisms are intertwiners. In particular, it provides the notion of Morita equivalence: two algebras $B,B'$ are Morita equivalent if there exist bimodules $_B M _{B'}$ and $_{B'} M' _B$ such that $M \otimes_{B'} M' \cong B$ as a $B$–$B$ bimodule and $M' \otimes_B M \cong B'$ as a $B'$–$B'$ bimodule. The unit object $\mathrm 1$ is, of course, an associative algebra. I will say that $B$ is Morita trivial if it is Morita equivalent to $\mathrm 1$.
Let me return now to the algebra $A = X^*\otimes X$. Is it Morita trivial? Note that $X$ is naturally a distinguished $\mathrm 1$–$A$ bimodule (i.e. right $A$-module), and $X^*$ is a distinguished $A$–$\mathrm 1$ bimodule. Moreover, it is almost trivial to check that $X^* \otimes_{\mathrm 1} X \cong A$ as an $A$–$A$ bimodule. The evaluation map moreover induces a homomorphism $X \otimes_A X^* \to \mathrm 1$ as a $\mathrm 1$–$\mathrm 1$ bimodule. If this induced homomorphism is an isomorphism, then yes, $A$ is Morita trivial.
Question: What are some examples of dualizable $X$ in categories $\mathcal C$ such that $A = X^* \otimes X$ is not Morita trivial?
Lest you think that there are no such examples, here's a trivial example. Let $\mathcal C$ be the category of $R$-modules for $R$ a commutative ring, and let $X = \mathrm 0$ be the zero $R$-module. Then $\mathrm 0$ is certainly dualizable, with dual $\mathrm 0^* = \mathrm 0$, and its endomorphism algebra is $\mathrm 0$, which is not Morita trivial unless $R$ is the zero ring.
On the other hand, here are some cases where $A$ is Morita trivial. If $X$ admits an endomorphism $f: X \to X$ with invertible trace then $A$ is Morita trivial — this is a fun exercise. This condition is sufficient but not necessary. For example, consider the category of representations over $\mathbb F_2$ of the cyclic group of order $3$. Let $X$ denote the unique $2$-dimensional irrep. Its dual is $X^* \cong X$. Since $X$ is an irrep, it has a unique non-zero endomorphism, which has trace $\dim X = 2 = 0$. Nevertheless, direct calculation shows that $A = X^* \otimes X$ is Morita-trivial (indeed, $X,X^*$ witness the Morita equivalence).
In general, I think that Morita-nontrivial endomorphism algebras determine "proper direct summands of $\mathcal C$ as a $\mathcal C$-module". I expect this means that if $R = S\oplus T$ is a direct sum of commutative rings, then $S$ and $T$ each determine Morita-nontrivial endomorphism algebras in the category of $R$-modules. Are there other examples?
