Rigid category
In
Definition
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 inverse is an object X−1 such that both X ⊗ X−1 and X−1 ⊗ X are isomorphic to 1, the identity object of the monoidal category. If an object X has a left (respectively right) inverse X−1 with respect to the tensor product then it is left (respectively right) rigid, and X* = X−1.
The operation of taking duals gives a
Uses
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
and its reciprocal isomorphism
.
Then for any endomorphism , the trace is of f is defined as the composition:
We may continue further and define the dimension of a rigid object to be:
.
Rigidity is also important because of its relation to 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.
Alternative terminology
A monoidal category where every object has a left (respectively right) dual is also sometimes called a left (respectively 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.
Discussion
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.
Notes
- ISBN 978-3-540-37477-0.
References
- Davydov, A. A. (1998). "Monoidal categories and functors". Journal of Mathematical Sciences. 88 (4): 458–472. .
- Rigid monoidal category at the nLab