Full and faithful functors


In category theory, a faithful functor is a functor that is injective when restricted to each set of morphisms that have a given source and target.

Formal definitions

Explicitly, let C and D be 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
for each X and Y in C.

Properties

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, and two morphisms f : X → Y and f′ : X′ → Y′ 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.
A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : C → D is a full and faithful functor and then.

Examples

The notion of a functor being 'full' or 'faithful' does not translate to the notion of a -category. In an -category, the maps between any two objects are given by a space only up to homotopy. Since the notion of injection and surjection are not homotopy invariant notions, we do not have the notion of a functor being "full" or "faithful." However, we can define a functor of quasi-categories to be fully faithful if for every X and Y in C, the map is a weak equivalence.