Information AboutForgetful Functor |
| CATEGORIES ABOUT FORGETFUL FUNCTOR | |
| functors | |
It is beneficial to distinguish between forgetful functors which "forget structure" versus those which "forget properties". For example, in the above example of commutative rings, in addition to those functors which delete some of the operations, there are functors which forget some of the axioms. There is a functor from the category CRng to '''Rng''' which forgets the axiom of commutativity, but keeps all the operations. Occasionally the object may include extra sets not defined strictly in terms of the underlying set (in this case, it is a matter of taste which part to consider the underlying set, though this is rarely ambiguous in practice). For these objects, there are forgetful functors which forget the extra sets which are more general. Most common objects studied in mathematics are constructed as underlying sets along with extra sets of structure on those sets (operations on the underlying set, privileged subsets of the underlying set, etc) which may satisfy some axioms. For these objects, a commonly considered forgetful functor is as follows. Let be any category based on Set s, e.g. Group s - sets of elements - or Topological Space s - sets of 'points'. As usual, write for the Object s of and write for the morphisms of the same. Consider the rule: | ||
| :<math>u</math> In <math>\operatorname{Fl}(\mathcal{C})\mapsto u | </math> the morphism, <math>u</math>, as a map of sets |
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| The Functor <math>\\</math> Is Then The Forgetful Functor From <math>\mathcal{C}</math> To <math>\mathbf{Set}</math>, The | "http://wwwinformationdelightinfo/encyclopedia/entry/category_of_sets" class="copylinks">Category Of Sets |
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