Quotient by an equivalence relation

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 R → f X × X → pr i X , i = 1 , 2 , {\displaystyle R\ {\overset {f}{\to }}\ X\times X\ {\overset {\operatorname {pr} _{i}}{\to }}\ X,\ \ i=1,2,} where R is an object in C and "f is an equivalence relation" means that, for any object T in C, the image (which is a set) of f : R ( T ) = Mor ⁡ ( T , R ) → X ( T ) × X ( T ) {\displaystyle f:R(T)=\operatorname {Mor} (T,R)\to X(T)\times X(T)} 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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