Let G be an algebraic group acting on a scheme X. Then f: X --> Y is a categorical quotient if it is constant on G-orbits and any other G-invariant morphism factors through it in a unique fashion. We say f is a 'good' categorical quotient if:
1) f is a surjective open submersion (i.e. the topology on Y is induced from X).
2) for any open U ⊂ Y, the induced map from functions on U to G-invariant functions on f^-1(U) is an isomorphism.
Does anyone know an example of a 'bad' categorical quotient (by which I mean...well...a not good one).