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In category theory, a branch of mathematics, the density theorem states that every presheaf of sets is a colimit of representable presheaves in a canonical way.
For example, by definition, a simplicial set is a presheaf on the simplex category Δ and a representable simplicial set is exactly of the form Δ n = Hom ] {\displaystyle \Delta ^{n}=\operatorname {Hom} ]} so the theorem says: for each simplicial set X,
where the colim runs over an index category determined by X.
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