Operad

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an operad is a structure that consists of abstract <a href="/facts/Operation_(mathematics)/Yplv602Q">operations</a>, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad 
  
    
      
        O
      
    
    {\displaystyle O}
  
, one defines an <a href="/facts/Operad_algebra/tHDxMG7R"><i>algebra over 
  
    
      
        O
      
    
    {\displaystyle O}
  
</i></a> to be a set together with concrete operations on this set which behave just like the abstract operations of 
  
    
      
        O
      
    
    {\displaystyle O}
  
. For instance, there is a <a href="/facts/Lie_operad/QchLFYR9">Lie operad</a> 
  
    
      
        L
      
    
    {\displaystyle L}
  
 such that the algebras over 
  
    
      
        L
      
    
    {\displaystyle L}
  
 are precisely the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a>; in a sense 
  
    
      
        L
      
    
    {\displaystyle L}
  
 abstractly encodes the operations that are common to all Lie algebras.  An operad is to its algebras as a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> is to its <a href="/facts/Group_representation/FsuuDagQ">group representations</a>.
</p>

Operad open-in-new

Operad