Full and faithful functors
- Full and faithful functors
In category theory, a faithful functor (resp. a full functor) is a functor which is injective (resp. surjective) when restricted to each set of morphisms with a given source and target.
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
*faithful if "F""X","Y" is injective
*full if "F""X","Y" is surjective
*fully faithful if "F""X","Y" is bijectivefor each "X" and "Y" 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 forgetful functor "U" : Grp → Set is faithful but is not injective on objects or on morphisms. This functor is not full as there are functions between groups which are not group homomorphisms. In general, a category with a faithful functor to Set is a concrete category: that forgetful functor is generally 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 is not surjective on objects or on morphisms.
* The forgetful functor Ab → Grp is fully faithful. It is injective on both objects and morphisms, but is surjective on neither.
ee also
*full subcategory
*equivalence of categories
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