In category theory, a functor between two categories $C,D$ is a map $F$ that assigns to each object (resp. morphism) $x$ of $C$ a corresponding object (resp. morphism) $F(x)$ of $D$ by respecting the incidence relations.For each pair of objects $x,y$ of $C$, we may then define a map $F_{x,y}$ that takes any morphism $m : x \rightarrow y$ to a morphism $F(m) : F(x) \rightarrow F(y)$. I understand that $F$ is called faithful if every such mapping $F_{x,y}$ is injective, which means intuitively that the relational structure of the category is preserved, although the objects may not be.There is a related notion of forgetful functor for which I couldn't find a precise definition, so is there anyone willing to help? I mean, is it just the opposite of faithful or is it the combination of unfaithfulness with some other implicit property?
Faithful functors vs forgetful functors: exact category-theoretic defs?
co.combinatorics;soft question;ct.category theory