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Tensor Category


TensorCategory1
TensorCategory2

In category theory, a tensor category (C, tensor ,I,a,r,l) consists of a category C, an object I of C, a functor  tensor :C×C->C, and a natural isomorphism

a=a_(UVW):(U tensor V) tensor W->U tensor (V tensor W)
(1)
r=r_V:V tensor I->V
(2)
l=l_V:I tensor V->V,
(3)

where the data are subject to the following axioms:

1. Given four objects U, V, W, and X of C, the top diagram above commutes.

2. Given two objects U and V of C, the bottom diagram above commutes.

In the above,  tensor is called the tensor product, a is called the associator, r is called the right unit, and l is called the left unit of the tensor category. The object I is referred to as the neutral element or the identity of the tensor product.

If the maps a, l, and r are always identities, the tensor category in question is said to be strict.

A related notion is that of a tensor R-category.


See also

Category, Category Theory, Functor, Morphism, Natural Isomorphism, Natural Transformation, Object, Strict Tensor Category, Unital Natural Transformation

This entry contributed by Christopher Stover

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References

Dieck, T. T. "Quantum Groups and Knot Algebra." 2000. http://www.uni-math.gwdg.de/tammo/dm.pdf.

Cite this as:

Stover, Christopher. "Tensor Category." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TensorCategory.html

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