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Given functors , a natural transformation involves components .
Is it necessary that these components already live in or can they be added as part of the definition of ?
Edit: I wasn't sure if I should put this in a query box on the nLab. I'm a little rusty.
Edit^2: For example,
Let consist of two objects and one morphism . Let consist of four objects and two morphisms , . We could have two functors
with and .
Would we just say there are no natural transformations or could we allow to bring along two morphisms , with it?
No. The natural transformation diagram has to commute. It's similar to the condition on a morphism of chain complexes.
@Sridhar: Thanks. That is what I thought the answer might be. Too bad.
I'm trying to flesh this out (to the best of meager abilities) at natural transformation (discussion). Help much appreciated! :) I'm probably on a dead end road, but by the time I discover that, I will hopefully have learned a thing or two. Basically, I was a little uncomfortable with the fact that natural transformations were defined only for functors sharing the same domain and codomain. In trying to see if I could cook up a slightly different natural transformation-like concept, I settled on the diagram there. I liked the idea enough to try to reduce the dimension by 1 and think of functors as special kind of natural transformation. Then maybe a morphism is a special kind of functor. Etc...
In a way, a cograph of a functor involves "bring(ing) along morphism" so I don't think the idea is entirely outlandish, I just need to find the right way to express it. Again, any ideas would be greatly appreciated.
Edit: There are also traces of discussion at Natural Transformation (ericforgy) and functor (discussion).
The reason that natural transformations have to share the same domain/codomain is that they're 2-morphisms in the category of categories. That is, we can assign to each pair of categories a functor category, and natural transformations are the morphisms in the functor category. I believe there are notions similar to what you're talking about using something called a double category, but I could be horribly wrong.
<p>Thanks Harry.</p>
<p>It's interesting you bring up the category of categories because that is one possible direction I was headed with this.</p>
<p>If natural transformations are 2-morphisms in the category of categories <a href="https://ncatlab.org/nlab/show/Cat">Cat</a>, i.e. bigons, what would be morphisms in the <a href="https://ncatlab.org/nlab/show/arrow+category">arrow category</a> of categories <img src="/extensions/vLaTeX/cache/latex_8b821b5ca328916550542edf98843025.png" title="Arr(Cat)" style="vertical-align:-20%;" class="tex" alt="Arr(Cat)" />. In <img src="/extensions/vLaTeX/cache/latex_8b821b5ca328916550542edf98843025.png" title="Arr(Cat)" style="vertical-align:-20%;" class="tex" alt="Arr(Cat)" />, unless I'm confused, objects would be functors but not necessarily with the same domain and codomain, i.e. we could have two objects</p>
<p><img src="/extensions/vLaTeX/cache/latex_733c874c86b3eec3e9369a1846cb8672.png" title="F:A\to B" style="vertical-align:-20%;" class="tex" alt="F:A\to B" /><br></p>
<img src="/extensions/vLaTeX/cache/latex_8467478d932a698e54905aa014665e65.png" title="G:C\to D" style="vertical-align:-20%;" class="tex" alt="G:C\to D" />
<p>where <img src="/extensions/vLaTeX/cache/latex_b754fe221a079c2ca5127d6ad18f886a.png" title="A" style="vertical-align:-20%;" class="tex" alt="A" />, <img src="/extensions/vLaTeX/cache/latex_525eb65281c6d3b0bfacfe7567f8af85.png" title="B" style="vertical-align:-20%;" class="tex" alt="B" />, <img src="/extensions/vLaTeX/cache/latex_ca43fb5496104dcafda44acbe4014b0e.png" title="C" style="vertical-align:-20%;" class="tex" alt="C" />, <img src="/extensions/vLaTeX/cache/latex_2741ee6cf4a5aa48b2ee29b3b0fee62e.png" title="D" style="vertical-align:-20%;" class="tex" alt="D" /> are distinct categories. What I am trying to describe (I think) would be morphisms between these. These morphisms would be commuting squares (of functors) like the one at <a href="https://ncatlab.org/nlab/show/natural+transformation+%28discussion%29">natural transformation (discussion)</a>.</p>
Ah yes, that's related to the "walking arrow", where you can compose 2-morphisms in a weird way by "messing around" with this walking arrow. The notion of a natural transformation that we have seems to be the right one, at least in the 2-category Cat.
I drew a diagram at Arrow Category of Categories (ericforgy). It is a little messy, but the point is there are three morphisms , , in the arrow category consisting of 4 objects
[Note: There are also composite morphisms and that I didn't draw.]
I think this is a little different than a double category because the 2-paths can branch as indicated. Does a double category allow this kind of "branching" of 2-paths?
I'm not sure I understand the diagram you drew. What are the codomains of and ?
Edit: Nevermind, I get it now; they'd be G and F, respectively.
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