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 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
for every pair of objects X and Y in C. The functor F is said to be
- faithful if FX,Y is injective
- full if FX,Y is surjective
- fully faithful if FX,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.
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
References
- Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (second ed.). Springer. ISBN 0-387-98403-8.
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