Density theorem (category theory)

In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a branch of mathematics, the density theorem states that every <a href="/facts/Presheaf_of_sets/gWTNAFo3">presheaf of sets</a> is a <a href="/facts/Colimit/JoLugLs3">colimit</a> of <a href="/facts/Representable_functor/dILwN4oj">representable presheaves</a> in a canonical way.
For example, by definition, a <a href="/facts/Simplicial_set/6DaqXMLT">simplicial set</a> is a presheaf on the simplex category Δ and a representable simplicial set is exactly of the form 
 
 
 
 
 Δ
 
 n
 
 
 =
 Hom
 ⁡
 (
 −
 ,
 [
 n
 ]
 )
 
 
 {\displaystyle \Delta ^{n}=\operatorname {Hom} (-,[n])}
 
 (called the standard n-simplex) so the theorem says: for each simplicial set X,

X
        ≃
        
          
            lim
            →
          
        
        ⁡
        
          Δ
          
            n
          
        
      
    
    {\displaystyle X\simeq \varinjlim \Delta ^{n}}

where the colim runs over an index category determined by X.

Density theorem (category theory) open-in-new

Density theorem (category theory)