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  Δ n = Hom ⁡ ( − , [ n ] ) {\displaystyle \Delta ^{n}=\operatorname {Hom} (-,[n])} (called the standard n-simplex) so the theorem says: for each simplicial set X,

X ≃ lim -> ⁡ Δ n {\displaystyle X\simeq \varinjlim \Delta ^{n}}

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

## Statement

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Let F be a presheaf on a category C; i.e., an object of the functor category C ^ = F c t ( C op , S e t ) {\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

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

It comes with the forgetful functor  p : I -> C {\displaystyle p:I\to C} .

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

I -> p C -> C ^ {\displaystyle I{\overset {p}{\to }}C\to {\widehat {C}}}

where the second arrow is the Yoneda embedding:  U ↦ h U = Hom ⁡ ( − , U ) {\displaystyle U\mapsto h_{U}=\operatorname {Hom} (-,U)} .

## Proof

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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:

Hom C ^ ⁡ ( F , G ) ≃ Hom ⁡ ( f , Δ G ) {\displaystyle \operatorname {Hom} _{\widehat {C}}(F,G)\simeq \operatorname {Hom} (f,\Delta _{G})}

where  Δ G {\displaystyle \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  lim -> − {\displaystyle \varinjlim -} is the left adjoint to the diagonal functor  Δ − . {\displaystyle \Delta _{-}.}

For this end, let  α : f -> Δ G {\displaystyle \alpha :f\to \Delta _{G}} be a natural transformation. It is a family of morphisms indexed by the objects in I:

α U , x : f ( U , x ) = h U -> Δ G ( U , x ) = G {\displaystyle \alpha _{U,x}:f(U,x)=h_{U}\to \Delta _{G}(U,x)=G}

that satisfies the property: for each morphism  ( U , x ) -> ( V , y ) , u : U -> V {\displaystyle (U,x)\to (V,y),u:U\to V} in I,  α V , y ∘ h u = α U , x {\displaystyle \alpha _{V,y}\circ h_{u}=\alpha _{U,x}} (since  f ( ( U , x ) -> ( V , y ) ) = h u . {\displaystyle f((U,x)\to (V,y))=h_{u}.} )

The Yoneda lemma says there is a natural bijection  G ( U ) ≃ Hom ⁡ ( h U , G ) {\displaystyle G(U)\simeq \operatorname {Hom} (h_{U},G)} . Under this bijection,  α U , x {\displaystyle \alpha _{U,x}} corresponds to a unique element  g U , x ∈ G ( U ) {\displaystyle g_{U,x}\in G(U)} . We have:

( G u ) ( g V , y ) = g U , x {\displaystyle (Gu)(g_{V,y})=g_{U,x}}

because, according to the Yoneda lemma,  G u : G ( V ) -> G ( U ) {\displaystyle Gu:G(V)\to G(U)} corresponds to  − ∘ h u : Hom ⁡ ( h V , G ) -> Hom ⁡ ( h U , G ) . {\displaystyle -\circ h_{u}:\operatorname {Hom} (h_{V},G)\to \operatorname {Hom} (h_{U},G).}

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

( G u ∘ θ V ) ( y ) = ( G u ) ( g V , y ) = g U , x = ( θ U ∘ F u ) ( y ) , {\displaystyle (Gu\circ \theta _{V})(y)=(Gu)(g_{V,y})=g_{U,x}=(\theta _{U}\circ Fu)(y),}

since  ( F u ) ( y ) = x {\displaystyle (Fu)(y)=x} . Clearly, the construction  α ↦ θ {\displaystyle \alpha \mapsto \theta } is reversible. Hence,  α ↦ θ {\displaystyle \alpha \mapsto \theta } is the requisite natural bijection.

## Notes

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1. ^ Mac Lane 1998, Ch III, § 7, Theorem 1.

## References

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