Example of natural transformation between covariant and contravariant functor
0
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It is OK to define a natural transformation between covariant and contravariant functor. But is there an example from geometry, topology or algebra?
category-theory
asked Jan 7 at 10:18
Guanyu LiGuanyu Li
144
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add a comment |
0
$begingroup$
It is OK to define a natural transformation between covariant and contravariant functor. But is there an example from geometry, topology or algebra?
category-theory
asked Jan 7 at 10:18
Guanyu LiGuanyu Li
144
$endgroup$
1
$begingroup$
You can define some notion of "transformation" between covariant and contravariant functors, but it is not the notion of a natural transformation. And even for more general notions such as extranatural or dinatural transformations, it makes more sense to think of them as between functors $mathcal C^{op}timesmathcal Ctomathcal D$. You can then specialize them by viewing a covariant/contravariant functor as a bifunctor that ignores the appropriate argument.
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– Derek Elkins
Jan 7 at 11:12
add a comment |
0
0
0
$begingroup$
It is OK to define a natural transformation between covariant and contravariant functor. But is there an example from geometry, topology or algebra?
category-theory
asked Jan 7 at 10:18
Guanyu LiGuanyu Li
144
$endgroup$
It is OK to define a natural transformation between covariant and contravariant functor. But is there an example from geometry, topology or algebra?
category-theory
category-theory
asked Jan 7 at 10:18
Guanyu LiGuanyu Li
144
asked Jan 7 at 10:18
Guanyu LiGuanyu Li
144
asked Jan 7 at 10:18
Guanyu LiGuanyu Li
144
asked Jan 7 at 10:18
Guanyu LiGuanyu Li
144
asked Jan 7 at 10:18
Guanyu LiGuanyu Li
144
144
1
$begingroup$
You can define some notion of "transformation" between covariant and contravariant functors, but it is not the notion of a natural transformation. And even for more general notions such as extranatural or dinatural transformations, it makes more sense to think of them as between functors $mathcal C^{op}timesmathcal Ctomathcal D$. You can then specialize them by viewing a covariant/contravariant functor as a bifunctor that ignores the appropriate argument.
$endgroup$
– Derek Elkins
Jan 7 at 11:12
add a comment |
1
$begingroup$
You can define some notion of "transformation" between covariant and contravariant functors, but it is not the notion of a natural transformation. And even for more general notions such as extranatural or dinatural transformations, it makes more sense to think of them as between functors $mathcal C^{op}timesmathcal Ctomathcal D$. You can then specialize them by viewing a covariant/contravariant functor as a bifunctor that ignores the appropriate argument.
$endgroup$
– Derek Elkins
Jan 7 at 11:12
1
1
$begingroup$
You can define some notion of "transformation" between covariant and contravariant functors, but it is not the notion of a natural transformation. And even for more general notions such as extranatural or dinatural transformations, it makes more sense to think of them as between functors $mathcal C^{op}timesmathcal Ctomathcal D$. You can then specialize them by viewing a covariant/contravariant functor as a bifunctor that ignores the appropriate argument.
$endgroup$
– Derek Elkins
Jan 7 at 11:12
$begingroup$
You can define some notion of "transformation" between covariant and contravariant functors, but it is not the notion of a natural transformation. And even for more general notions such as extranatural or dinatural transformations, it makes more sense to think of them as between functors $mathcal C^{op}timesmathcal Ctomathcal D$. You can then specialize them by viewing a covariant/contravariant functor as a bifunctor that ignores the appropriate argument.
$endgroup$
– Derek Elkins
Jan 7 at 11:12
add a comment |
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1
$begingroup$
You can define some notion of "transformation" between covariant and contravariant functors, but it is not the notion of a natural transformation. And even for more general notions such as extranatural or dinatural transformations, it makes more sense to think of them as between functors $mathcal C^{op}timesmathcal Ctomathcal D$. You can then specialize them by viewing a covariant/contravariant functor as a bifunctor that ignores the appropriate argument.
$endgroup$
– Derek Elkins
Jan 7 at 11:12