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
For two objects, we write for the hom-object.
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, in that
where denotes the Yoneda embedding. In module-language, using the tensor product of functors, this reads
Yet another way to state this is as a colimit over the comma category , for the Yoneda embedding:
This statement we call the co-Yoneda lemma.
To show that a presheaf is canonically presented as a colimit of representables, we exhibit a natural isomorphism
By the definition of the 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
Let Set, and recall the definition of the coend as a coequalizer
This says that the coend is the set of equivalence classes of pairs
where two such pairs
are regarded as equivalent if there exists
(Because then the two pairs are the two images of the pair under the two morphisms being coequalized.)
But now considering the case that and , so that shows that any pair
is identified, in the coequalizer, with the pair
hence with , and that this coequalizing operation is the action
of morphisms on elements of the presheaf by pullback.
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.
For enrichment over (pointd compactly generated topological spaces) the co-Yoneda lemma in the sense of every presheaf is a colimit of representables appears for instance as
The co-Yoneda lemma in the sense of MacLane appears as a brief uncommented exercise on p. 63 of
where it is atributed to Daniel Kan.