Full functor
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In category theory, a full functor is a functor which is surjective when restricted to each set of morphisms with a given source and target.
In other words, a functor F : C → D is full if the maps
- FX,Y : MorC(X, Y) → MorD(FX, FY)
are surjective for every pair of objects X and Y in C.
Note that a full functor need not be surjective on objects or morphisms. That is, 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.
For example, 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.
Another example is the forgetful functor Ab → Grp. This is full, but neither surjective on objects or morphisms. A counterexample is the forgetful functor Grp → Set. This is not full as there are functions between group which are not group homomorphisms.
See also: