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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.{{harvnb|Mac Lane|1998|loc=Ch III, § 7, Theorem 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 \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

Let F be a presheaf on a category C; i.e., an object of the functor category \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) 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 \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{Hom} (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, x}: 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, 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 \operatorname{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: \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, 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:

:(G u \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.

Notes

{{reflist}}

References

Category:Representable functors