Dual object

Summary

A dual of an object $A$ in a monoidal category is an object $A^*$ with an evaluation $A^* \otimes A \to I$ and a coevaluation $I \to A \otimes A^*$. These two maps must satisfy the zigzag identities. It abstracts the relation between a finite-dimensional vector space and its dual space. A dual, when it exists, is unique up to unique isomorphism. Duals give traces and dimensions in monoidal categories and correspond to bending strings in string diagrams.