Density theorem (category theory)

In , 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 (called the standard n-simplex) so the theorem says: for each simplicial set X, where the colim runs over an index category determined by X. Let F be a presheaf on a category C; i.e., an object of the . For an index category over which a colimit will run, let I be the of F: it is the category where an object is a pair consisting of an object U in C and an element , a morphism consists of a morphism in C such that It comes with the forgetful functor . Then F is the colimit of the (i.e., a functor) where the second arrow is the Yoneda embedding: . Let f denote the above diagram. To show the colimit of f is F, we need to show: for every presheaf G on C, there is a natural bijection: where is the constant functor with value G and Hom on the right means the set of natural transformations. This is because the universal property of a colimit amounts to saying is the left adjoint to the diagonal functor For this end, let be a natural transformation. It is a family of morphisms indexed by the objects in I: that satisfies the property: for each morphism in I, (since ) The Yoneda lemma says there is a natural bijection . Under this bijection, corresponds to a unique element . We have: because, according to the Yoneda lemma, corresponds to Now, for each object U in C, let be the function given by . This determines the natural transformation ; indeed, for each morphism in I, we have: since . Clearly, the construction is reversible. Hence, is the requisite natural bijection.