Bowling average

In category theory, a faithful functor (resp. a full functor) is a functor that is injective (resp. surjective) when restricted to each set of morphisms that have 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

$F_{X,Y}:{\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{𝒞}}(X,Y)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{𝒟}}(F(X),F(Y))$

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 bijective

for 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 (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) 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.

A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : C → D is a full and faithful functor and $F(x)\cong F(y)$ then $x\cong y$.

Examples

• The forgetful functor U : Grp → Set is faithful as each group maps to a unique set and the group homomorphism are a subset of the functions. This functor is not full as there are functions between groups which are not group homomorphisms. A category with a faithful functor to Set is (by definition) a concrete category; in general, that forgetful functor is not full.

• Let F : Set → Set be the functor which maps every set to the empty set and every function to the empty function. Then F is full, but is neither injective on objects nor on morphisms.

• The inclusion functor Ab → Grp is fully faithful.

See also

• full subcategory
• equivalence of categories

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534