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    • CommentRowNumber1.
    • CommentAuthorEric
    • CommentTimeNov 5th 2009

    Asked a question at natural transformation.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2009

    I added a quick reply.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2009

    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.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeNov 5th 2009
    • CommentRowNumber5.
    • CommentAuthorEric
    • CommentTimeNov 5th 2009

    I proposed a definition for something at natural transformation. What is a good (or existing) name for it?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2009

    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.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019
    • (edited Apr 3rd 2019)

    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.

    diff, v51, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019

    Okay, fixed.

  1. Thanks!

  2. Add a title to the alternative definition of natural transformation and their composition in terms of arrowwise components.

    Luidnel Maignan

    diff, v54, current

    • CommentRowNumber11.
    • CommentAuthorJem Lord
    • CommentTimeNov 5th 2020

    Alternative condition for morphismwise definition and horizontal composition in terms of morphismwise definition.

    diff, v56, current

    • CommentRowNumber12.
    • CommentAuthorHurkyl
    • CommentTimeNov 7th 2020

    Typo fix. Changed “B C:=[B,C]B^C := [B,C]” to “C B:=[B,C]C^B := [B,C]”.

    diff, v57, current

  3. Replace a diagram with tikzcd


    diff, v58, current

  4. added text from HoTT book


    diff, v59, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2022

    Please, what you just added is the standard classical definition. This has nothing to do with HoTT. The edit should be reverted.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2022

    have reverted the edit from #14 (see also the discussion here)

    diff, v60, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2023

    added pointer to Borceux, here and in related entries

    diff, v61, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023

    added pointer to:

    diff, v64, current

  5. Created example section for natural transformations and added some. Maybe someone wants to add more.

    diff, v67, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2024
    • (edited Feb 3rd 2024)


    I have now added the original example (here) which motivated Eilenberg & MacLane 1945 to introduce the notion of natural transformation in the first place.

    diff, v68, current

    • CommentRowNumber21.
    • CommentAuthorRodMcGuire
    • CommentTimeFeb 4th 2024

    general example

    For a category CC its category of presheaves Set C opSet^{C^op} has for objects all functors C opSetC^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.

    diff, v69, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2024
    • (edited Feb 4th 2024)

    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 CC 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?

    diff, v70, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2024
    • (edited Feb 4th 2024)

    Ah, probably you mean the case where CC is the category {EV}\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…

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2024
    • (edited Feb 4th 2024)

    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

    diff, v71, current

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2024
    • (edited Feb 4th 2024)

    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.

    diff, v71, current