Let C and C‘ be two abelian categories and let F:C---gt; C‘ a (additive?) functor. Suppose that X non zero implies F(X) non zero. Does this imply that F is faithfull?
Yes, as long as there isn't any mud or other stuff on the bb. I wouldn't try to clean them.
The functor C → D is one which maps every object of C to a fixed object X in D and every morphism in C to the identity morphism on X. Such a functor is called a constant or selection functor. Maybe you can take CC' and the functor to be the constant functor. For example CAb, the category of abelian groups and the functor that sends a group to say, (Z,+). That would show that it's not necessarily to have a faithfull functor. The forgetfull functors are faithfull though.
F need not be faithful (even if it's additive). consider the functor F:C --- C given by F(X) X, F(f) f for all f in Hom(X,X) and F(g) 0 for all g in Hom(X,Y) for X not equal to Y. this functor satisfies your condition that X nonzero implies F(X) nonzero trivially. also, F is additive since it induces the identity homomorphism on Hom(X,X) and the zero homomorphism on Hom(X,Y). so, provided that C has two distinct nonzero objects X and Y with at least one nonzero morphism in Hom(X,Y), we have an additive functor which is not faithful.
of course they can be fired twice, if you can find them ;-)
