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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

• CommentRowNumber3.
• CommentAuthorHurkyl
• CommentTimeMay 18th 2022

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

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

• CommentRowNumber4.
• CommentAuthorHurkyl
• CommentTimeMay 18th 2022

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

• 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.

• 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.