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Full and faithful functors - Examples | | Full and faithful functors - Examples: Encyclopedia II - Full and faithful functors - Examples | | The forgetful functor U : Grp → Set is faithful but neither injective on objects or morphisms. This functor is not full as there are functions between groups which are not group homomorphisms. 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 i ...
See also:Full and faithful functors, Full and faithful functors - Examples | | | Full and faithful functors, Full and faithful functors - Examples, full subcategory, equivalence of categories | | |
| | | Full and faithful functors: Encyclopedia II - Full and faithful functors - Examples
Full and faithful functors - Examples
The forgetful functor U : Grp → Set is faithful but neither injective on objects or morphisms. This functor is not full as there are functions between groups which are not group homomorphisms. 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.
Other related archivesCategory theory, bijective, categories, category theory, concrete category, empty function, empty set, equivalence of categories, forgetful functor, full subcategory, functor, group homomorphisms, groups, injective, locally small, morphisms, surjective
 Adapted from the Wikipedia article "Examples", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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