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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.
added pointer to:
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 $C$?
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.
Could you be more specific about which part of the entry you are referring to?
Added a cross-reference to topological concrete category.
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