Density theorem (category theory)
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
- 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 .
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
References
- Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. 5 (2nd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.