Dual object

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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.

Contents 1 Definition 2 Categories with duals 3 See also 4 References

[edit] Definition

In a monoidal category , a pair of dual objects is a pair (A,B) of objects together with two morphisms

and

such that

and

.

In this situation, the object A is called a left dual of B, and B is called a right dual of A. 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.

[edit] Categories with duals

A monoidal category where every object has a left (resp. right) dual is sometimes called a left (resp. right) autonomous 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.

[edit] See also

• Dualizing object

[edit] References

• Peter Freyd and David Yetter (1989). "Braided Compact Closed Categories with Applications to Low-Dimensional Topology". Advances in Mathematics 77: 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.