Quotient by an equivalence relation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, given a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> C, a quotient of an <a href="/facts/Object_(category_theory)/7WzxUThf">object</a> X by an equivalence relation 
 
 
 
 f
 :
 R
 →
 X
 ×
 X
 
 
 {\displaystyle f:R\to X\times X}
 
 is a <a href="/facts/Coequalizer/UjkYMAfm">coequalizer</a> 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 <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>) 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 <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a>; that is, a <a href="/facts/Reflexive_relation/NzTytEg9">reflexive</a>, <a href="/facts/Symmetric_relation/ekCWDIAH">symmetric</a> and <a href="/facts/Transitive_relation/9r90y50r">transitive</a> <a href="/facts/Binary_relation/r9BwGgaK">relation</a>.
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 <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaves</a>.

Quotient by an equivalence relation open-in-new

Quotient by an equivalence relation