F-coalgebra

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically in <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, an 
  
    
      
        F
      
    
    {\displaystyle F}
  
-coalgebra is a <a href="/facts/Mathematical_structure/6CXReUNv">structure</a> defined according to a <a href="/facts/Functor/l2u3rIBE">functor</a> 
  
    
      
        F
      
    
    {\displaystyle F}
  
, with specific properties as defined below. For both <a href="/facts/Algebraic_structure/TkuXca24">algebras</a> and <a href="/facts/Coalgebra/pvx52xS6">coalgebras</a>, a functor is a convenient and general way of organizing a <a href="/facts/Signature_(logic)/C0YQCuce">signature</a>.  This has applications in <a href="/facts/Computer_science/lN3pEyPz">computer science</a>: examples of coalgebras include <a href="/facts/Lazy_evaluation/mD3DxDkc">lazy evaluation</a>, <a href="/facts/Recursion_(computer_science)/2uXo18Gv">infinite</a> <a href="/facts/Data_structure/yNsNvL1I">data structures</a>, such as <a href="/facts/Stream_(computing)/2jtBAp0U">streams</a>, and also <a href="/facts/State_transition_system/AABVmNzn">transition systems</a>.
</p><p>
  
    
      
        F
      
    
    {\displaystyle F}
  
-coalgebras are <a href="/facts/Dual_(category_theory)/frtUrhBq">dual</a> to <a href="/facts/F-algebra/qH3vvtzw">
  
    
      
        F
      
    
    {\displaystyle F}
  
-algebras</a>.  Just as the class of all <a href="/facts/Algebraic_structure/TkuXca24">algebras</a> for a given signature and equational theory form a <a href="/facts/Variety_(universal_algebra)/vXWYFyDI">variety</a>, so does the class of all 
  
    
      
        F
      
    
    {\displaystyle F}
  
-coalgebras satisfying a given equational theory form a covariety, where the signature is given by 
  
    
      
        F
      
    
    {\displaystyle F}
  
.
</p>

F-coalgebra open-in-new

F-coalgebra