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.
]]>So I have made the mentioning of presheaf categories a remark (now here) right after the existing remark on functor categories, in the Definition section
and then expanded out the comment on graph homomorphisms to an example about diagram homomorphisms, here
]]>Ah, probably you mean the case where is the category so that its category of (co)presheaves is that of directed graphs with graph homomorphisms?
I am editing to clarify this…
]]>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 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?
]]>general example
]]>For a category its category of presheaves has for objects all functors 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.
Great.
I have now added the original example (here) which motivated Eilenberg & MacLane 1945 to introduce the notion of natural transformation in the first place.
]]>Created example section for natural transformations and added some. Maybe someone wants to add more.
]]>added pointer to:
added pointer to Borceux, here and in related entries
]]>have reverted the edit from #14 (see also the discussion here)
]]>Please, what you just added is the standard classical definition. This has nothing to do with HoTT. The edit should be reverted.
]]>added text from HoTT book
Anonymous
]]>Replace a diagram with tikzcd
Anonymous
]]>Typo fix. Changed “” to “”.
]]>Alternative condition for morphismwise definition and horizontal composition in terms of morphismwise definition.
]]>Add a title to the alternative definition of natural transformation and their composition in terms of arrowwise components.
Luidnel Maignan
]]>Thanks!
]]>Okay, fixed.
]]>Somebody alerts me that the first diagram here does not come out right. At least for me here on this phone. But we can just have a proper LaTeX implementation now anyway. Unless somebody is quicker, I’ll do it a little later.
]]>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.
]]>I proposed a definition for something at natural transformation. What is a good (or existing) name for it?
]]>Also at ericforgy:Natural Transformation.
]]>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.
]]>I added a quick reply.
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