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
(called the standard n-simplex) so the theorem says: for each simplicial set X,
![{\displaystyle X\simeq \varinjlim \Delta ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/609ab852f3a1af564c3e78d15c23a44a0dc0487d)
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
. 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
consisting of an object U in C and an element
,
- a morphism
consists of a morphism
in C such that ![{\displaystyle (Fu)(y)=x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8fa1694fefa5c60416e3a6304ed301d81e5e21c)
It comes with the forgetful functor
.
Then F is the colimit of the
diagram (i.e., a functor)
![{\displaystyle I{\overset {p}{\to }}C\to {\widehat {C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faf0d6e498579d120d47265883ffc25574612532)
where the second arrow is the
Yoneda embedding:
.
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:
![{\displaystyle \operatorname {Hom} _{\widehat {C}}(F,G)\simeq \operatorname {Hom} (f,\Delta _{G})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/592f7f7bd99bc0f67d42430b5e855a9ea869b07c)
where
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
is the left adjoint to the diagonal functor
For this end, let
be a natural transformation. It is a family of morphisms indexed by the objects in I:
![{\displaystyle \alpha _{U,x}:f(U,x)=h_{U}\to \Delta _{G}(U,x)=G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ededc2adc0732c21cee693e9fe92d21d5169ea92)
that satisfies the property: for each morphism
in I,
(since
)
The Yoneda lemma says there is a natural bijection
. Under this bijection,
corresponds to a unique element
. We have:
![{\displaystyle (Gu)(g_{V,y})=g_{U,x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b31306a01acfc6767f4daae5ec929292a68f685)
because, according to the Yoneda lemma,
corresponds to
Now, for each object U in C, let
be the function given by
. This determines the natural transformation
; indeed, for each morphism
in I, we have:
![{\displaystyle (Gu\circ \theta _{V})(y)=(Gu)(g_{V,y})=g_{U,x}=(\theta _{U}\circ Fu)(y),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e7a3d22218c2ddcc4c23b1223f8c3b63e9c26d)
since
. Clearly, the construction
is reversible. Hence,
is the requisite natural bijection.
Notes
References