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In mathematics, given a category C, a quotient of an object X by an equivalence relation f : R → X × X {\displaystyle f:R\to X\times X} is a coequalizer for the pair of maps
where R is an object in C and "f is an equivalence relation" means that, for any object T in C, the image of f : R = Mor → X × X {\displaystyle f:R=\operatorname {Mor} \to X\times X} is an equivalence relation; that is, a reflexive, symmetric and transitive relation.
The basic case in practice is when C is the category of all schemes over some scheme S. But the notion is flexible and one can also take C to be the category of sheaves.
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