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 FigueroaO'Farrill Aug 9 '10 at 20:11
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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. 

