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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 18th 2011

    Added the statement of the Isbell-Freyd characterization of concrete categories, in the special case of finitely complete categories for which it looks more familiar, along with the proof of necessity.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2021

    added pointer to:

    diff, v28, current

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeMay 18th 2022

    I corrected the article to list Set Set_\bot (sets and partial functions) as an example of a a concrete category.

    Is RelRel not concretizable? The explanation given is inadequate; it only suggests a specific functor is not faithful. (and… I’m not actually sure what the suggested functor is intended to be)

    The article also lists Rel(C)Rel(C) as a nonexample, but that’s incorrect, since every small category is concretizable. Was there supposed to be an additional condition given on CC?

    diff, v33, current

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeMay 18th 2022

    Added to examples the case of small categories, and of locally small categories with small separators.

    diff, v33, current

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022
    I think they meant "internal relation" in place of "congruence" when defining Rel(C).
    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeMay 18th 2022

    Rel is also concretizable, since the singleton set is a separator, so I’ve moved it to the examples section.

    I think the original intent was to provide examples where there was a naive notion of “underlying set” that did not extend to a functor. So I’ve added some additional language to the “examples” section to that effect.

    diff, v36, current

    • CommentRowNumber7.
    • CommentAuthorHurkyl
    • CommentTimeMay 18th 2022

    Oh, and I’ve also added the example of the “prime spectrum” on the opposite category of rings as a nonexample (it’s not faithful), and listed Ho(Top) in the nonexamples section as well.

    diff, v36, current

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