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,

satisfying

$\lambda_X\circ(\varepsilon\otimes \mathrm{id}_X)\circ\alpha_{X,X^*,X}^{-1}\circ(\mathrm{id}_X\otimes\eta)\circ\rho_X^{-1}=\mathrm{id}_X$

and

$\rho_{X^*}\circ(\mathrm{id}_{X^*}\otimes\varepsilon)\circ\alpha_{X^*,X,X^*}\circ(\eta\otimes \mathrm{id}_{X^*})\circ\lambda_{X^*}^{-1}=\mathrm{id}_{X^*}$.

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.

See also

• Dualizing object

References

• Peter Freyd and David Yetter (1989). "Braided Compact Closed Categories with Applications to Low-Dimensional Topology". Advances in Mathematics 77 (2): 156–182. doi:10.1016/0001-8708(89)90018-2. 
• André Joyal and Ross Street. "The Geometry of Tensor calculus II". Synthese Lib 259: 29–68.