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 concept of duals in the sense of finite-dimensional vector spaces. A consequence of these axioms is that 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? At the Research School on Bicategories, Categorification and Quantum Theory, Heunen suggested that this is not the case. In this note, we will prove his claim by constructing an explicit counterexample.
|Publication status||Published - 9 Jan 2023|
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- math.CT, math.QA, 18D15, 18M10