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Unital Natural Transformation


UnitalNaturalTransformation1
UnitalNaturalTransformation2

A natural transformation Phi_Y:B(AY)->Y is called unital if the leftmost diagram above commutes. Similarly, a natural transformation Psi_Y:Y->A(BY) is called unital if the diagram on the right-hand side above commutes.

Note that in these definitions, A, B, and Y are all objects in a tensor category C, I is the neutral (or identity) object in C, and the juxtaposition AB is shorthand for the tensor product A×B in C. What's more, the subscripts attached to the transformations Phi={Phi_Y} and Psi={Psi_Y} denote the components of the functors (indexed with respect to the objects in C) in question.


See also

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

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. "Unital Natural Transformation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/UnitalNaturalTransformation.html

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