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A concrete category is a category C together with a function that assigns to each object A of C a set called the underlying set of A.
Example: The category of groups, equipped with the function that assigns to each group its underlying set in the usual sense, is a concrete category. What is the underlying set for an object in a category of groups ?

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closed as no longer relevant by Jonas Meyer, Robin Chapman, Qiaochu Yuan, Pete L. Clark, José Figueroa-O'Farrill Aug 9 '10 at 20:11

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As far as I can tell you answered your own question just before you asked it. – Jonas Meyer Aug 9 '10 at 19:33
This question is probably better suited to - mathoverflow is intended for “research level” questions, whereas this is more like an exercise for a first course in category theory. – Peter LeFanu Lumsdaine Aug 9 '10 at 19:34
You don't want a function, but a functor, and it has to be faithful. – Qiaochu Yuan Aug 9 '10 at 19:39

The underlying set for an object in the category of groups is just the underlying set of the group, there is no other name for it.

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