Talk:Compact closed category

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The definition here is not the usual definition of trace. Trace of f:A --> A should be a morphism I to I, where I is the unit for the monoidal structure. Think in Vect, where Vect(k,k) \cong k. Then tr(f) = e_A c (f # 1) n_A, where n and e are the coevaluation and evaluation maps, c is the braiding or symmetry, and "#" is the tensor product. This gives the usual notion of trace.

For traces in categories that are not compact, you need a pivotal structure.

Also, adjoint are of course only defined for morphisms. In order to say that an object is a left adjoint, you need to mention that you are looking at a monoidal category as a one-object bicategory. There is obviously no need for that, because you can define a left dual as you do.

Should say, adjoints are defined for functors, so if you want to say A^* is the left adjoint for A then it is better to say A^* \otimes - is the left adjoint of A \otimes -. Or indeed, explain the one object bicategory version. —Preceding unsigned comment added by 60.241.132.115 (talk) 11:12, 6 December 2007 (UTC)