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.

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 ${\mathrm{\Delta }}^n=\mathrm{Hom}(-,[n])$ (called the standard n-simplex) so the theorem says: for each simplicial set X,

$X\simeq \underrightarrow{\lim }{\mathrm{\Delta }}^n$

where the colim runs over an index category determined by X.

Statement

Let F be a presheaf on a category C; i.e., an object of the functor category $\hat{C}=𝐅𝐜𝐭(C^{\text{op}},𝐒𝐞𝐭)$. For an index category over which a colimit will run, let I be the category of elements of F: it is the category where

1. an object is a pair $(U,x)$ consisting of an object U in C and an element $x\in F(U)$,
2. a morphism $(U,x)\to (V,y)$ consists of a morphism $u:U\to V$ in C such that $(Fu)(y)=x.$

It comes with the forgetful functor $p:I\to C$.

Then F is the colimit of the diagram (i.e., a functor)

$I\stackrel{p}{\to }C\to \hat{C}$

where the second arrow is the Yoneda embedding: $U\mapsto h_U=\mathrm{Hom}(-,U)$.

Proof

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:

${\mathrm{Hom}}_{\hat{C}}(F,G)\simeq \mathrm{Hom}(f,{\mathrm{\Delta }}_G)$

where ${\mathrm{\Delta }}_G$ 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 $\underrightarrow{\lim }-$ is the left adjoint to the diagonal functor ${\mathrm{\Delta }}_{-}.$

For this end, let $\alpha :f\to {\mathrm{\Delta }}_G$ be a natural transformation. It is a family of morphisms indexed by the objects in I:

${\alpha }_{U,x}:f(U,x)=h_U\to {\mathrm{\Delta }}_G(U,x)=G$

that satisfies the property: for each morphism $(U,x)\to (V,y),u:U\to V$ in I, ${\alpha }_{V,y}\circ h_u={\alpha }_{U,x}$ (since $f((U,x)\to (V,y))=h_u.$)

The Yoneda lemma says there is a natural bijection $G(U)\simeq \mathrm{Hom}(h_U,G)$. Under this bijection, ${\alpha }_{U,x}$ corresponds to a unique element $g_{U,x}\in G(U)$. We have:

$(Gu)(g_{V,y})=g_{U,x}$

because, according to the Yoneda lemma, $Gu:G(V)\to G(U)$ corresponds to $-\circ h_u:\mathrm{Hom}(h_V,G)\to \mathrm{Hom}(h_U,G).$

Now, for each object U in C, let ${\theta }_U:F(U)\to G(U)$ be the function given by ${\theta }_U(x)=g_{U,x}$. This determines the natural transformation $\theta :F\to G$; indeed, for each morphism $(U,x)\to (V,y),u:U\to V$ in I, we have:

$(Gu\circ {\theta }_V)(y)=(Gu)(g_{V,y})=g_{U,x}=({\theta }_U\circ Fu)(y),$

since $(Fu)(y)=x$. Clearly, the construction $\alpha \mapsto \theta$ is reversible. Hence, $\alpha \mapsto \theta$ is the requisite natural bijection.

References

• Template:Cite book