Duality in Monoidal Categories

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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.


Original languageEnglish
Publication statusPublished - 9 Jan 2023
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  • math.CT, math.QA, 18D15, 18M10