| Faithful Functor |
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| CATEGORIES ABOUT FULL AND FAITHFUL FUNCTORS | |
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Explicitly, let ''C'' and ''D'' be ( Locally Small ) Categories and let ''F'' : ''C'' → ''D'' be a functor from ''C'' to ''D''. The functor ''F'' induces a function : for every pair of objects ''X'' and ''Y'' in ''C''. The functor ''F'' is said to be
for each ''X'' and ''Y'' in in ''C''. A faithful functor need not be injective on objects or morphisms. That is, two objects ''X'' and ''X''′ may map to the same object in ''D'', and two morphisms ''f'' : ''X'' → ''Y'' and ''f''′ : ''X''′ → ''Y''′ may map to the same morphism in ''D''. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in ''D'' not of the form ''FX'' for some ''X'' in ''C''. Morphisms between such objects clearly cannot come from morphisms in ''C''. EXAMPLES The s which are not Group Homomorphism s. More generally, for any Concrete Category the forgetful functor to Set is faithful (but usually not full). Let ''F'' : ''C'' → Set be the functor which maps every object in ''C'' to the Empty Set and every morphism to the Empty Function . Then ''F'' is full, but neither surjective on objects or morphisms. The forgetful functor Ab → '''Grp''' is fully faithful. However, it is neither surjective on objects or morphisms. SEE ALSO |
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