In higher category theory
Cohomology and homotopy
In higher category theory
What is sometimes called the co-Yoneda lemma is a basic fact about presheaves (a basic fact of topos theory): it says that every presheaf is a colimit of representables and more precisely that it is the “colimit over itself of all the representables contained in it”.
One might think of this as related by duality to the Yoneda lemma, hence the name.
Every presheaf is a colimit of representables
Recall that the Yoneda lemma says that for a -enriched category, a -valued presheaf on and an object of , there is a natural isomorphism in
Using the definition of the enriched functor category on the left in terms of an end, this reads
In this form the Yoneda lemma is also referred to as Yoneda reduction.
Under abstract duality an end turns into a coend, so a co-Yoneda lemma ought to be a similarly fundamental expression for in terms of a coend.
The natural candidate is the statement that every presheaf is a colimit of representables which may be stated as
where denotes the Yoneda embedding. In module language, using the tensor product of functors, this reads
This statement we call the co-Yoneda lemma.
Yet another way to state this is as a colimit over the comma category , for the Yoneda embedding:
To show that a presheaf is canonically presented as a colimit of representables, we exhibit a natural isomorphism
By the definition of coend, maps are in natural bijection with families of maps extranatural in and natural in . Those are in natural bijection with families of maps natural in and extranatural in . These are in natural bijection with families of maps (natural in ), where the isomorphism is by the Yoneda lemma. Thus we have exhibited a natural isomorphism
(natural in ). By Yoneda again, this gives .
If one follows the Yoneda-lemma argument at the end, one arrives at the explicit isomorphism
Namely, it corresponds to the family of maps
(extranatural in and natural in ) which in turn corresponds to the natural family
associated with the structure of the functor .
MacLane’s co-Yoneda lemma
In a brief uncommented exercise on MacLane, p. 62
the following statement, which is atrributed to Kan, is called the co-Yoneda lemma.
For a category, Set the category of sets, a functor, let be the comma category of elements , let be the projection and let for each the functor be the diagonal functor sending everything to the constant value .
The co-Yoneda lemma in the sense of Kan/MacLane is the statement that there is a natural isomorphism of functor categories
Here is an outline of an explicit proof:
A natural transformation assigns to each element an element , i.e., an arrow . We define a corresponding transformation which assigns to each object in the morphism . It is easy to check that the naturality condition on corresponds to the naturality condition on , and that the correspondence is bijective.
Here is a more conceptual proof in terms of comma categories:
Set classifies discrete fibrations, in the sense that a functor classifies the discrete fibration
and natural transformations correspond to maps of fibrations
i.e. functor which commute on the nose with the projections , to the base category ).
This applies in particular to . Notice the category of elements is the co-slice , with its usual projection to .
However, the comma category is the “lax pullback” (really, the comma object, the discussion at 2-limit) appearing in
and so a fibration map corresponds exactly to a lax square
This yields the co-Yoneda lemma in the sense of MacLane’s exercise.
The coYoneda lemma appears as a brief uncommented exercise on p. 63 of
where it is atributed to Kan.
A blog discussion which led to the creation of this entry is here.