*-autonomous category

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In mathematics, a *-autonomous category C is a symmetric monoidal closed category equipped with a dualizing object $\bot$ .

More explicitly, in every symmetric monoidal closed category C, for every objects A and $\bot$ , there exists a morphism

$\partial_{A,\bot}:A\to(A\Rightarrow\bot)\Rightarrow\bot$

defined as the image by the bijection defining the monoidal closure, of the morphism

$\mathrm{eval}_{A,A\Rightarrow\bot}\circ\gamma_{A\Rightarrow\bot,A} : (A\Rightarrow\bot)\otimes A\to\bot$

An object $\bot$ of the category C is called dualizing when the associated morphism $\partial_{A,\bot}$ is an isomorphism for every object A of the category C.

Equivalently, a *-autonomous category is a symmetric monoidal closed category C together with a functor $(-)^\bot:C^{\mathrm{op}}\to C$ such that for every object A there is a natural isomorphism $A\cong{A^\bot}^\bot$ , and for every three objects A, B and C there is a natural bijection

$\mathrm{Hom}(A\otimes B,C^\bot)\cong\mathrm{Hom}(A,(B\otimes C)^\bot)$ .

The dualizing object of C is then defined by $\bot=1^\bot$ .

[edit] References

Michael Barr (1979). ".*-autonomous Categories". Lecture Notes in Mathematics 752.
Michael Barr (1995). "Non-symmetric *-autonomous Categories". Theoretical Computer Science 139.

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Category: Monoidal categories

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*-autonomous category

From Wikipedia, the free encyclopedia

[edit] References

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