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:
“I see you boys of summer in your ruin.
Man in his maggot’s barren.
And boys are full and foreign in the pouch.
I am the man your father was.
We are the sons of flint and pitch.
O see the poles are kissing as they cross.”
—Dylan Thomas (1914–1953)
“Farewell? a long farewell to all my greatness.
This is the state of man; today he puts forth
The tender leaves of hopes, tomorrow blossoms,
And bears his blushing honors thick upon him:
The third day comes a frost, a killing frost,
And when he thinks, good easy man, full surely
His greatness is a-ripening, nips his root,
And then he falls as I do.”
—William Shakespeare (1564–1616)
“Well meant are the wounds a friend inflicts, but profuse are the kisses of an enemy.”
—Bible: Hebrew, Proverbs 27:6.
KJ translation reads: Faithful are the wounds of a friend.