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Statement

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

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

• 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.

References

1. Mac Lane 1998, Ch III, § 7, Theorem 1. - 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. https://zbmath.org/?format=complete&q=an:0906.18001 ↩