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.

[edit] Definition

In a monoidal category $(\mathbf{C},\otimes,I)$ , a pair of dual objects is a pair (A,B) of objects together with two morphisms

$\eta_A:I\to B\otimes A$ and $\varepsilon_A:A\otimes B\to I$

such that

$\lambda_A\circ(\varepsilon_A\otimes A)\circ\alpha_{A,B,A}^{-1}\circ(A\otimes\eta_A)\circ\rho_A^{-1}=\mathrm{id}_A$

and

$\rho_{B}\circ(B\otimes\varepsilon_A)\circ\alpha_{B,A,B}\circ(\eta_A\otimes B)\circ\lambda_{B}^{-1}=\mathrm{id}_{B}$ .

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] 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.