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.
Let F be a presheaf on a category C; i.e., an object of the functor category
{\displaystyle {\widehat {C}}=\mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}
For an index category over which a colimit will run, let I be the category of elements of F: it is the category where It comes with the forgetful functor
Then F is the colimit of the diagram (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
α : f →
be a natural transformation.
It is a family of morphisms indexed by the objects in I: that satisfies the property: for each morphism
α
α
) The Yoneda lemma says there is a natural bijection
Under this bijection,
α
corresponds to a unique element
We have: because, according to the Yoneda lemma,
corresponds to
be the function given by
This determines the natural transformation
α ↦ θ
α ↦ θ
is the requisite natural bijection.