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).

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I don't think you mean open immersion. – David Zureick-Brown Oct 16 '09 at 21:06
Correct; it's sub- not im- – Harold Williams Oct 16 '09 at 22:52