# Dual object

In category theory, a branch of mathematics, it is possible to define a concept of dual object generalizing the concept of dual space in linear algebra.

A category in which each object has a dual is called autonomous or rigid.

## Definition

Consider an object $X$ in a monoidal category $(\mathbf{C},\otimes, I, \alpha, \lambda, \rho)$. The object $X^*$ is called a left dual of $X$ if there exist two morphsims

$\eta:I\to X\otimes X^*$, called the coevaluation, and $\varepsilon:X^*\otimes X\to I$, called the evaluation,

such that the following two diagrams commute

 and

The object $X$ is called the right dual of $X^*$. Left duals are canonically isomorphic when they exist, as are right duals. When C is braided (or symmetric), every left dual is also a right dual, and vice versa.

If we consider a monoidal category as a bicategory with one object, a dual pair is exactly an adjoint pair.

## Categories with duals

A monoidal category where every object has a left (resp. right) dual is sometimes called a left (resp. right) autonomous category. Algebraic geometers call it a left (resp. right) rigid category. A monoidal category where every object has both a left and a right dual is called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.