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/or faithful:
“To judge the appearances we receive of things, we should need a judicatory instrument; to verify this instrument, we should need a demonstration; to rectify this demonstration, we should need an instrument: so here we are arguing in a circle.
Seeing the senses cannot decide our dispute, being themselves full of uncertainty, we must have recourse to Reason; there is no reason but must be built upon another reason, so here we are retreating backwards to all eternity.”
—Michel de Montaigne (1533–1592)
“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.