Density theorem (category theory) explained

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

\Delta ^{n} = \operatorname{Hom}( - , [n]) (called the standard n-simplex) so the theorem says: for each simplicial set X,X \simeq \varinjlim \Delta ^{n} where the colim runs over an index category determined by X.

Statement

\widehat{C} = Fct (C ^{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) ,

1. 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 \overset{p}\to C \to \widehat{C} where the second arrow is the Yoneda embedding: 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:

\operatorname{Hom} _{\widehat{C} } (F, G) \simeq \operatorname (f, \Delta_G)where \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 \varinjlim - is the left adjoint to the diagonal functor \Delta _{-} .

For this end, let

\alpha: f \to \Delta _{G} be a natural transformation. It is a family of morphisms indexed by the objects in I:\alpha _{U,} : f(U, x) = h _{U} \to \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,} \circ h _{u} = \alpha _{U,} (since f((U, x) \to (V, y)) = h _{u.} )

The Yoneda lemma says there is a natural bijection

G(U) \simeq \operatorname{Hom}(h _{U,} G) . Under this bijection, \alpha _{U,} corresponds to a unique element g _{U,} \in G(U) . We have:(Gu)(g _{V,} ) = g _{U,} because, according to the Yoneda lemma, Gu: G(V) \to G(U) corresponds to - \circ h _{u:} \operatorname{Hom}(h _{V,} G) \to \operatorname{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,} . 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:(G u \circ \theta _{V)(y)} = (Gu)(g _{V,} ) = g _{U,} = (\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

• Book: Mac Lane, Saunders . Saunders Mac Lane . . 2nd . . 5 . New York, NY . . 1998 . 0-387-98403-8 . 0906.18001 .