---
tag: 8A7YXMW727
kind: concept
name: Dual object
msc: [18M05, 18M10]
sources:
  - url: https://en.wikipedia.org/wiki/Dual_object
same_as:
  wikidata: Q5310222
  wikipedia: Dual object
---

## Summary

A dual of an object $A$ in a [[monoidal category|M3MSQ3NKAE]] 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|07JBSZE0V4]].
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.

## Explanations

### L1 human

A finite-dimensional vector space $V$ has a dual $V^*$ of linear functionals.
One can pair a functional with a vector to get a number, and one can produce a canonical tensor $\sum_i e_i \otimes e^i$ out of nothing.
Doing one after the other, in the right order, gives back the identity: these are the zigzag identities.
A dual object is an object with these two maps and these two identities, in any monoidal category.
In pictures, the two maps are a cap and a cup, and the identities say that a zigzag in a string can be pulled straight.