Dual object

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.


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

Dual-one.png and Dual-two.png

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[edit]

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[edit]


  • 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.