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Natural Transformation


Let F,G:C->D be functors between categories C and D. A natural transformation Phi from F to G consists of a family Phi_C:F(C)->G(C) of morphisms in D which are indexed by the objects C of C so that, for each morphism f:C->D between objects in C, the equality

 G(f) degreesPhi_C=Phi_D degreesF(f):F(C)->G(D)

holds. The elements Phi_C are called the components of the natural transformation.

If all the components Phi_C are isomorphisms in D, then Phi is called a natural isomorphism between F and G. In this case, one writes Phi:F=G.


See also

Category, Category Theory, Functor, Morphism, Natural Isomorphism, Object, 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. "Natural Transformation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/NaturalTransformation.html

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