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Asked a question at natural transformation.
I added a quick reply.
Todd has indicated the characterization of natural transformations in terms of the cartesian closed monoidal structure on Cat in the discussion section.
I have now created a subsection in the main body of the entry on this perspective.
Also at ericforgy:Natural Transformation.
I proposed a definition for something at natural transformation. What is a good (or existing) name for it?
replied further in the discussion there.
I think we are talking here really about the operation that takes a 2-category to a double category. We should draw the relevant diagrams at double category, eventually.
Okay, fixed.
Thanks!
Please, what you just added is the standard classical definition. This has nothing to do with HoTT. The edit should be reverted.
added pointer to:
general example
For a category $C$ its category of presheaves $Set^{C^op}$ has for objects all functors $C^op \rightarrow Set$ and for morphisms all natural transformations between those presheaves. If the objects in that category can be seen as being some type of graph then its morphisms are graph homomorphisms which in this case are natural transformations.
To the presheaf example I added a handful more words and hyperlinks.
(By the way, this is not really an example, more an elaboration of the definition. I would suggest to move this to right after the main definition.)
I don’t understand what the currently last sentence of the example is trying to say
(“If the objects of $C$ can be seen as being some type of graph then its morphisms are graph homomorphisms which in this case are natural transformations.”)
You mean to consider presheaves on a category of graphs?
Ah, probably you mean the case where $C$ is the category $\big\{E \rightrightarrows V\big\}$ so that its category of (co)presheaves is that of directed graphs with graph homomorphisms?
I am editing to clarify this…
I have deleted the content of the Properties section, due to revision 59 by the notorious “Anonymous” (cf. #14 and following):
The deleted paragraphs consisted of nothing but trivial re-iteration of the definition, with wording that alluded to HoTT but in standard math prose with no specific connection to HoTT.
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