Last week, I started a series on sieves and Grothendieck topoi. This is a short intermezzo on a well-known lemma from category theory that I will need for next week’s instalment.
Let be a small category. Recall that a presheaf on
is a functor
. Examples include the representable presheaves
for
, given by
. The Yoneda lemma says that for any presheaf
and any
, the map
Given a functor , write
for the comma category whose objects are pairs
where
and
is a morphism (natural transformation) in
. A morphism
is a morphism
such that the triangle
(1)
By the Yoneda lemma, the category is isomorphic (not just equivalent!) to the category
of pairs
with
and
with morphisms
given by morphisms
in
such that
. It’s convenient to keep both points of view.
Lemma. Let be a small category, and let
be a functor.
- The object
is naturally a cocone under
via the morphisms
.
- This cocone makes
the colimit of the diagram
of representable functors.
In particular, any presheaf on a small category is a colimit of representable presheaves.
Proof. A cocone under is a presheaf
with a natural transformation
to the constant diagram
with value
. This means every
is taken to a natural transformation
, such that for any morphism
, the square
(1) To make a cocone under
, simply take
.
(2) Given any other cocone under
, define the natural transformation
by
One can also easily rewrite this argument in terms of natural transformations . For instance, the universal cocone
is the natural transformation
of functors
given on
by
. Naturality of
follows at once from (1). But checking that this thing is universal is a bit more tedious in this language.
Example. A standard example where this point of view is useful is simplicial sets. Let be the category of finite nonempty totally ordered sets, with (weakly) monotone increasing functions as morphisms. A simplicial set is a functor
, and we often think of them as combinatorial models for topological spaces. The representable ones are the standard
-simplices
, where
is the totally ordered set
for
.
If is a simplicial set, its value at
is called the
-simplices
of
. By the Yoneda lemma, this is
. Then the story above is saying that a simplicial set is the colimit over all its
-simplices for
. This is extremely useful, as many arguments proceed by attaching simplices one at a time.