underlying set

Given a concrete category, that is a category CC equipped with a functor UU from CC to the category of sets (satisfying certain conditions), we call UU a forgetful functor and call U(x)U(x), for xx an object of CC, the underlying set of xx.

In the case where CC is explicitly a category of structured sets, then every object xx of CC is a set |x|{|x|} equipped with some extra structure. In that case, the underlying set of xx is precisely this set |x|{|x|}.

More generally, if CC is concrete over DD, so we have U:CDU\colon C \to D faithful, then given any object x:Cx\colon C, its underlying object of DD is U(x)U(x).

Last revised on August 29, 2012 at 19:41:41. See the history of this page for a list of all contributions to it.