Abelian 2-group

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an Abelian 2-group is a higher dimensional analogue of an <a href="/facts/Abelian_group/FfWwmx4R">Abelian group</a>, in the sense of <a href="/facts/Higher-dimensional_algebra/vyMHqRp4">higher algebra</a>, which were originally introduced by <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a> while studying abstract structures surrounding <a href="/facts/Abelian_varieties/dGSDVruG">Abelian varieties</a> and <a href="/facts/Picard_group/lWH0CoWQ">Picard groups</a>. More concretely, they are given by <a href="/facts/Groupoid/hvj8wvWe">groupoids</a> 
  
    
      
        
          A
        
      
    
    {\displaystyle \mathbb {A} }
  
 which have a <a href="/facts/Bifunctor/l2u3rIBE">bifunctor</a> 
  
    
      
        +
        :
        
          A
        
        ×
        
          A
        
        →
        
          A
        
      
    
    {\displaystyle +:\mathbb {A} \times \mathbb {A} \to \mathbb {A} }
  
 which acts formally like the addition an Abelian group. Namely, the bifunctor 
  
    
      
        +
      
    
    {\displaystyle +}
  
 has a notion of <a href="/facts/Commutativity/WaYxJLxd">commutativity</a>, <a href="/facts/Associativity/EQX8lV6r">associativity</a>, and an <a href="/facts/Identity_element/9nss3xHY">identity</a> structure. Although this seems like a rather lofty and <a href="/facts/Abstract_structure/EkSkcwdQ">abstract structure</a>, there are several (very concrete) examples of Abelian 2-groups. In fact, some of which provide prototypes for more complex examples of higher algebraic structures, such as Abelian <a href="/facts/N-group_(category_theory)/h2XalQ26"><i>n</i>-groups</a>.
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Abelian 2-group open-in-new

Abelian 2-group