# Difference between revisions of "Natural transformation"

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A natural transformation is a way of turning one [[functor]] into another functor while 'preserving' the structure of the [[Category (category theory)|categories]]. Natural transformations can be thought of a 'morphisms between functors,' and indeed they are precisely the morphisms in functor categories. | A natural transformation is a way of turning one [[functor]] into another functor while 'preserving' the structure of the [[Category (category theory)|categories]]. Natural transformations can be thought of a 'morphisms between functors,' and indeed they are precisely the morphisms in functor categories. | ||

− | More precisely, given two categories <math>\mathcal{C}</math> and <math>\mathcal{D}</math>, and two functors <math>F,G:\mathcal{C}\to \mathcal{D}</math>, then a natural transformation <math>\varphi:F\to G</math> is a mapping which assigns to each object <math>X\in \text{Ob}(\mathcal{C})</math> a morphism <math>\varphi_X:F(X)\to G(X)</math> in <math>\mathcal{D}</math> such that for every morphism <math>f:X\to Y</math> of <math>\mathcal{C}</math>, we have:<cmath>\varphi_Y\circ F(f) = G(f)\circ \varphi_X.</cmath> | + | More precisely, given two categories <math>\mathcal{C}</math> and <math>\mathcal{D}</math>, and two functors <math>F,G:\mathcal{C}\to \mathcal{D}</math>, then a natural transformation <math>\varphi:F\to G</math> is a mapping which assigns to each object <math>X\in \text{Ob}(\mathcal{C})</math> a morphism <math>\varphi_X:F(X)\to G(X)</math> in <math>\mathcal{D}</math> such that for every morphism <math>f:X\to Y</math> of <math>\mathcal{C}</math>, we have:<cmath>\varphi_Y\circ F(f) = G(f)\circ \varphi_X.</cmath>This equation can also be expressed by saying that the following diagram [[Commutative diagram|commutes]]: |

+ | <asy> | ||

+ | draw((1.5,0)--(8.5,0),EndArrow); | ||

+ | draw((1.5,10)--(8.5,10),EndArrow); | ||

+ | draw((0,9)--(0,1),EndArrow); | ||

+ | draw((10,9)--(10,1),EndArrow); | ||

+ | |||

+ | label("$G(f)$",(5,-1)); | ||

+ | label("$F(f)$",(5,11)); | ||

+ | label("$\varphi_X$",(-1,5)); | ||

+ | label("$\varphi_Y$",(11,5)); | ||

+ | |||

+ | label("$G(X)$",(0,0)); | ||

+ | label("$G(Y)$",(10,0)); | ||

+ | label("$F(X)$",(0,10)); | ||

+ | label("$F(Y)$",(10,10)); | ||

+ | </asy> | ||

{{stub}} | {{stub}} | ||

[[Category:Category theory]] | [[Category:Category theory]] |

## Revision as of 21:15, 2 September 2008

A natural transformation is a way of turning one functor into another functor while 'preserving' the structure of the categories. Natural transformations can be thought of a 'morphisms between functors,' and indeed they are precisely the morphisms in functor categories.

More precisely, given two categories and , and two functors , then a natural transformation is a mapping which assigns to each object a morphism in such that for every morphism of , we have:This equation can also be expressed by saying that the following diagram commutes:

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