Density theorem (category theory)

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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.[1]

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.

Proof

Let F be a presheaf on a category C; i.e., an object of the functor category . As an index category over which a colimit will run, let I be the category of elements of F: it is the category where

  1. an object is a pair consisting of an object U in C and an element ,
  2. a morphism consists of a morphism in C such that

It comes with the forgetful functor .

Now, we claim F is the colimit of the diagram (i.e., a functor)

where the second arrow is the Yoneda embedding: . Let f denote this 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.

Notes

  1. ^ Mac Lane, Ch III, § 7, Theorem 1.

References

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