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 \Delta^n = \operatorname(-, (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 $\widehat = \mathbf(C^\text, \mathbf)$. For an index category over which a colimit will run, let ''I'' be the category of elements of ''F'': it is the category where
# an object is a pair $(U, x)$ consisting of an object ''U'' in ''C'' and an element $x \in F(U)$,
# 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)
:$I \overset\to C \to \widehat$
where the second arrow is the Yoneda embedding: $U \mapsto h_U = \operatorname(-, U)$.

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:

:$\operatorname_ (F, G) \simeq \operatorname (f, \Delta_G)$
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'':
:$\alpha_: f(U, x) = h_U \to \Delta_G(U, x) = G$
that satisfies the property: for each morphism $(U, x) \to (V, y), u: U \to V$ in ''I'', $\alpha_ \circ h_u = \alpha_$
(since $f((U, x) \to (V, y)) = h_u.$)

The Yoneda lemma says there is a natural bijection $G(U) \simeq \operatorname(h_U, G)$. Under this bijection, $\alpha_$ corresponds to a unique element $g_ \in G(U)$. We have:
:$(Gu)(g_) = g_$
because, according to the Yoneda lemma, $Gu: G(V) \to G(U)$ corresponds to $- \circ h_u: \operatorname(h_V, G) \to \operatorname(h_U, G).$

Now, for each object ''U'' in ''C'', let $\theta_U: F(U) \to G(U)$ be the function given by $\theta_U(x) = g_$. 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:
:$(G u \circ \theta_V)(y) = (Gu)(g_) = g_ = (\theta_U \circ Fu)(y),$
since $(Fu)(y) = x$. Clearly, the construction $\alpha \mapsto \theta$ is reversible. Hence, $\alpha \mapsto \theta$ is the requisite natural bijection.

Notes

References

* {{cite book , last=Mac Lane , first=Saunders , author-link=Saunders Mac Lane , title=Categories for the Working Mathematician , edition=2nd , series=Graduate Texts in Mathematics , volume=5 , location=New York, NY , publisher=Springer-Verlag , year=1998 , isbn=0-387-98403-8 , zbl=0906.18001

Representable functors