Compact closed category

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In category theory, a symmetric monoidal category $(\mathbf{C},\otimes,I)$ is compact closed when to every object $A$ there is an assigned left adjoint, that is an object $A *$ called the dual of $A$ together with two arrows $\eta_A:I\to A^*\otimes A$ and $\varepsilon_A:A\otimes A^*\to I$ such that

$\lambda_A\circ(\varepsilon_A\otimes A)\circ\alpha_{A,A^*,A}^{-1}\circ(A\otimes\eta_A)\circ\rho_A^{-1}=\mathrm{id}_A$

and

$\rho_{A^*}\circ(A^*\otimes\varepsilon_A)\circ\alpha_{A^*,A,A^*}\circ(\eta_A\otimes A^*)\circ\lambda_{A^*}^{-1}=\mathrm{id}_{A^*}$ .

[edit] Properties

Every compact closed category C admits a trace. Namely, for every morphism $f:A\otimes C\to B\otimes C$ , one can define

$\mathrm{Tr_{A,B}^C}(f)=\rho_B\circ(B\otimes\varepsilon_C)\circ\alpha_{B,C,C^*}\circ(f\otimes C^*)\circ\alpha_{A,C,C^*}^{-1}\circ(A\otimes\eta_{C^*})\circ\rho_B^{-1}:A\to B$