Are all Categorical Quotient a GIT quotient?
in the sense that if $U$ is a subset of $X$ such that the categorical quotient $U//G$ exists, there is a linearizion $L$ such that $U$ is in $X^{ss}$ (the semi stable locus of $X$ with respect to $L$) and $U//G$ is naturally a subset of $X^{ss}//G$.
Dolgachev in his "Lecture on Invariant Theory" suggested that could be the case while introduction rational quotient, however did not actually prove it.