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  "body": "\n## Definition\n\nA *rigid monoidal category* is a [[monoidal category|M3MSQ3NKAE]] $\\mathcal{C}$ in which every object $A$ has both a right [[dual|8A7YXMW727]] $A^*$ and a left dual ${}^*A$.\nThat is, there is an object $A^*$ with morphisms $A^* \\otimes A \\to I$ and $I \\to A \\otimes A^*$ satisfying the zigzag identities.\nThere is also an object ${}^*A$ with morphisms $A \\otimes {}^*A \\to I$ and $I \\to {}^*A \\otimes A$ satisfying the zigzag identities.\n\n## Remarks\n\nExample: the category $\\mathbf{FinVect}_k$ of finite-dimensional vector spaces over a field $k$ is rigid.\nFor $V$ both duals can be taken to be the dual space $V^*$, with evaluation and the coevaluation $1 \\mapsto \\sum_i e_i \\otimes e^i$ or its flip.\nNon-example: the category $\\mathbf{Vect}_k$ of all vector spaces is not rigid, because an infinite-dimensional space has no dual.\nIn a rigid category, a morphism $f \\colon A \\to B$ has a transpose $f^* \\colon B^* \\to A^*$, and $A \\mapsto A^*$ becomes a contravariant functor.\nIn a braided monoidal category, an object has a right dual if and only if it has a left dual, so asking for one kind suffices.\nRigid monoidal categories are also called autonomous categories; a symmetric rigid monoidal category is called compact closed.\n",
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