Categorical Quotients

AbstractEvery equivalence relation R on an algebraic variety U defines a class PR of all morphisms constant on equivalence classes of R and determines its categorical closure R̄ defined on U by xR̄y if and only if φ(x)=φ(y), for every φ∈PR. It is proved (Theorem A) that the equivalence classes of R̄ coincide with fibers of a morphism ψ∈PR. In the family of all morphisms with this property we may determine a subfamily of morphisms, called final pseudoquotients, which contains categorical quotients, if a categorical quotient exists, and, in the general case, is a substitute of such quotients