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| Definition | |
| 2>
There are at least two equivalent definitions of a rigidity.
An object X of a monoidal category is called left rigid if there is an object Y and morphisms and such that both compositions
are identities. A right rigid object is defined similarly.
An object X of a monoidal category is called left rigid if it has a left dual X* := [X, 1] and a morphism such that the compositions
[citation needed]
are identities. The definition of a right rigid object holds mutatis mutandis.
An inverse is an object X-1 such that both X ⊗ X-1 and X-1 ⊗ X are isomorphic to 1, the one object of the monoidal category. If an object X has a left (resp. right) inverse X-1 with respect to the tensor product then it is left (resp. right) rigid, and X* = X-1.
The operation of taking duals gives a contravariant functor on a rigid category.
[edit] Tags:Monoidal Category,Dual, | |
| Uses | |
| 2>
One important application of rigidity is in the definition of the trace of an endomorphism of a rigid object. The trace can be defined for any rigid category such that taking the ( )**, the functor of taking the dual twice repeated, is isomorphic to the identity functor. Then for any right rigid object X, and any other object Y, we may define the isomorphism
.
Then for any endomorphism , the trace is of f is defined as the composition
,
so . We may continue further and define the dimension of a rigid object to be
.
Rigidity is also importance because of its relation internal Hom's. If X is a left rigid object, then every internal Hom of the form [X, Z] exists and is isomorphic to Z ⊗ Y. In particular, in a rigid category, all internal Hom's exist.
[edit] Tags: | |
| Alternative Terminology | |
| 2>
A monoidal category where every object has a left (resp. right) dual is also sometimes called a left (resp. right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.
[edit] Tags:Autonomous Category,Symmetric,Compact Closed Category, | |
| Notes | |
| 2>
^ Dold, A.; Puppe, D. (1980). "Duality, trace and transfer". Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), (PWN): 81–102.
[edit] Tags: | |
| References | |
| 2>
Davydov, A. A. (1998). "Monoidal categories and functors". Journal of Mathematical Sciences 88 (4): 458–472. doi:10.1007/BF02365309. http://www.springerlink.com/content/kv27r034rn00x48m.
[edit] Tags:Categories,Monoidal Categories, | |
| See also | |
| 2>
A monoidal category is a category with a tensor product, precisely the sort of category for which rigidity makes sense.
The category of pure motives is formed by rigidifying the category of effective pure motives.
Retrieved from "http://en.wikipedia.org/w/index.php?title=Rigid_category&oldid=443890367"
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