- #1
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Still working on wrapping my head around these things in general...
It strikes me that, unless I'm really flubbing something up, a natural isomorphism T between two functors F,G:A-->B can be regarded itself as a functor B-->B.
For the objects, we just send each object X to the target of the morphism [itex]T_X[/itex].
For a morphism f:X-->Y, we have:
[tex]
Tf := T(X) \stackrel{T_X^{-1}}{\longrightarrow}
X \stackrel{f}{\longrightarrow}
Y \stackrel{T_Y}{\longrightarrow}
T(Y)
[/tex]
So this appears to be a functor... then, of course we have that the evaluation T(F) is simply given by T o F as functors.
As I write this I realize that this can't be a useful interpretation of natural transformations in general, because I had to use the fact the isomorphisms have inverses...
But is it a useful way to think of natural isomorphisms? Or am I just going to mislead and confuse myself and should stop thinking about this immediately?
It strikes me that, unless I'm really flubbing something up, a natural isomorphism T between two functors F,G:A-->B can be regarded itself as a functor B-->B.
For the objects, we just send each object X to the target of the morphism [itex]T_X[/itex].
For a morphism f:X-->Y, we have:
[tex]
Tf := T(X) \stackrel{T_X^{-1}}{\longrightarrow}
X \stackrel{f}{\longrightarrow}
Y \stackrel{T_Y}{\longrightarrow}
T(Y)
[/tex]
So this appears to be a functor... then, of course we have that the evaluation T(F) is simply given by T o F as functors.
As I write this I realize that this can't be a useful interpretation of natural transformations in general, because I had to use the fact the isomorphisms have inverses...
But is it a useful way to think of natural isomorphisms? Or am I just going to mislead and confuse myself and should stop thinking about this immediately?