Difference between revisions of "Natural transformation"
(New page: A natural transformation is a way of turning one functor into another functor while 'preserving' the structure of the categories. Natural transformations...) |
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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:![\[\varphi_Y\circ F(f) = G(f)\circ \varphi_X.\]](http://latex.artofproblemsolving.com/2/4/0/240d371946d89b12baf5bf4eb56cc383a25648dd.png)
This equation can also be expressed by saying that the following 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]](http://latex.artofproblemsolving.com/4/a/1/4a1291f5f2886e17b816ed45ca09d63e742e515c.png)
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