[<< wikibooks] Homological Algebra/Abelian Category
== Exercises ==
Given 
  
    
      
        f
        :
        a
        →
        b
      
    
    {\displaystyle f:a\to b}
   in an 
  
    
      
        A
        b
      
    
    {\displaystyle Ab}
  -enriched category with zero object. Prove that 
  
    
      
        f
        =
        
          0
          
            a
            ,
            b
          
        
      
    
    {\displaystyle f=0_{a,b}}
   iff 
  
    
      
        f
      
    
    {\displaystyle f}
   factors through 
  
    
      
        
          0
        
      
    
    {\displaystyle \mathbf {0} }
  .Given a biproduct 
  
    
      
        (
        a
        ⊕
        b
        ,
        
          i
          
            1
          
        
        ,
        
          p
          
            1
          
        
        ,
        
          i
          
            2
          
        
        ,
        
          p
          
            2
          
        
        )
      
    
    {\displaystyle (a\oplus b,i_{1},p_{1},i_{2},p_{2})}
   of 
  
    
      
        a
      
    
    {\displaystyle a}
   and 
  
    
      
        b
      
    
    {\displaystyle b}
  . Prove that 
  
    
      
        (
        a
        ⊕
        b
        ,
        
          i
          
            1
          
        
        ,
        
          i
          
            2
          
        
        )
      
    
    {\displaystyle (a\oplus b,i_{1},i_{2})}
   is a coproduct of 
  
    
      
        a
      
    
    {\displaystyle a}
   and 
  
    
      
        b
      
    
    {\displaystyle b}
   and 
  
    
      
        (
        a
        ⊕
        b
        ,
        
          p
          
            1
          
        
        ,
        
          p
          
            2
          
        
        )
      
    
    {\displaystyle (a\oplus b,p_{1},p_{2})}
   is a product of 
  
    
      
        a
      
    
    {\displaystyle a}
   and 
  
    
      
        b
      
    
    {\displaystyle b}
  .In an 
  
    
      
        A
        b
      
    
    {\displaystyle Ab}
  -enriched category with zero object, a kernel of 
  
    
      
        f
        :
        a
        →
        b
      
    
    {\displaystyle f:a\to b}
   can be equivalently be characterized as a pullback of 
  
    
      
        
          0
        
        →
        b
      
    
    {\displaystyle \mathbf {0} \to b}
   along 
  
    
      
        f
      
    
    {\displaystyle f}
  .