Quasi-projective reduction of toric varieties

We define a quasi-projective reduction of a complex algebraic variety X to be a regular map from X to a quasi-projective variety that is universal with respect to regular maps from X to quasi-projective varieties. A toric quasi-projective reduction is the analogous notion in the category of toric varieties, For a given toric variety X we first construct a toric quasi-projective reduction. Then we show that X has a quasi-projective reduction if and only if its toric quasi-projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi-projective toric variety admits a categorical quotient in the category of quasi-projective varieties.