Is there a special name or any research on Cartesian compact closed categories?

Question

As per the title. I can't find anything about the combination of the two, and such categories interest me. Does anyone know of any such categories?

Answer 1

The only dualisable object in a cartesian monoidal category is the terminal object. Thus, a cartesian compact closed category must be trivial.

Indeed, suppose $A$ has a dual $A^*$. Since the unit object is terminal, the counit $\epsilon : A \times A^* \to 1$ is forced. Consider the unit $\eta : 1 \to A^* \times A$. This decomposes into components as a morphism $1 \to A^*$ and a morphism $1 \to A$. By definition, $(\epsilon \times \mathrm{id}_A) \circ (\mathrm{id}_A \times \eta) = \mathrm{id}_A$; but $\epsilon \times \mathrm{id}_A : A \times A^* \times A \to A$ is the third projection, so the condition means that $A \to 1 \to A$ is $\mathrm{id}_A$. On the other hand, $1 \to A \to 1$ must be $\mathrm{id}_1$, so we deduce $A \cong 1$.