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.
Read more about Full And Faithful Functors: Examples
Famous quotes containing the words full and, full and/or faithful:
“Much of a man’s character will be found betokened in his backbone. I would rather feel your spine than your skull, whoever you are. A thin joist of a spine never yet upheld a full and noble soul.”
—Herman Melville (1819–1891)
“The woods were as fresh and full of vegetable life as a lichen in wet weather, and contained many interesting plants; but unless they are of white pine, they are treated with as little respect here as a mildew, and in the other case they are only the more quickly cut down.”
—Henry David Thoreau (1817–1862)
“We only part to meet again.
Change, as ye list, ye winds: my heart shall be
The faithful compass that still points to thee.”
—John Gay (1685–1732)