Duality in Monoidal Categories

Abstract: We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal-hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the latter, the internal-hom functor is implemented by tensoring with the respective duals. This raises the question: can one decide whether a closed monoidal category is rigid, simply by verifying that the internal-hom is tensor-representable? We provide a counterexample in terms of finitely-generated projective objects in an abelian k-linear category. A byproduct of our work is that we obtain characterisations of the Grothendieck--Verdier duality, also called *-autonomy, and rigidity of functor categories endowed with Day convolution as their tensor product. Applied to Mackey functors, this yields a proof of a sketched argument by Bouc linking rigidity of an object to it being finitely-generated projective.