In higher category theory
The Yoneda lemma says that the set of morphisms from a representable presheaf into an arbitrary presheaf is in natural bijection with the set assigned by to the representing object .
The Yoneda lemma is an elementary but deep and central result in category theory and in particular in sheaf and topos theory. It is essential background behind the central concepts of representable functor, universal construction, and universal element.
Recall that for a locally small category and the category of presheaves on , there naturally is a functor
– called the Yoneda embedding for reasons explained below – which sends to the category of presheaves over it: this is just the image of the Hom-functor
under the Hom-adjunction
in the closed symmetric monoidal category of categories.
Hence sends any object to the presheaf which assigns to any other object of the set of morphisms from into :
One way to appreciate the meaning of this and of what the Yoneda lemma has to say about it is to regard this in the context of space and quantity: thinking of the objects of as test spaces, presheaves on are generalized spaces modeled on which are characterized by the way one can map objects of into them.
The Yoneda lemma states that the functor has good properties which make this interpretation consistent.
The Yoneda Lemma
Let be a locally small category, the category of presheaves on . Let be an object.
The Yoneda lemma asserts that the set of morphisms from the presheaf represented by into any other presheaf is in natural bijection with the set that this presheaf assigns to .
There is a canonical isomorphism
natural in .
Here denotes the functor category, also denoted and the representable presheaf. This is the standard notation used mostly in pure category theory and enriched category theory. In other parts of the literature it is customary to denote the presheaf represented by as . In that case the above is often written
to emphasize that the morphisms of presheaves are natural transformations of the corresponding functors.
The proof is by chasing the element around both legs of a naturality square for a transformation :
What this diagram shows is that the entire transformation is completely determined from the single value , because for each object of , the component must take an element (i.e., a morphism ) to , according to the commutativity of this diagram.
The crucial point is that the naturality condition on any natural transformation is sufficient to ensure that is already entirely fixed by the value of its component on the identity morphism . And every such value extends to a natural transformation .
More in detail, the bijection is established by the map
where the first step is taking the component of a natural transformation at and the second step is evaluation at .
The inverse of this map takes to the natural transformation with components
In the light of the interpretation in terms of space and quantity mentioned above this says that for a generalized space modeled on , and for a test space, morphisms from to with regarded as a generalized space are just the morphisms from into .
The Yoneda lemma has the following direct consequences. As the Yoneda lemma itself, these are as easily established as they are useful and important.
corollary I: Yoneda embedding
The Yoneda lemma implies that the Yoneda embedding functor really is an embedding in that it is a full and faithful functor, because for it naturally induces the isomorphism of Hom-sets.
corollary II: uniqueness of representing objects
Since the Yoneda embedding is a full and faithful functor, an isomorphism of representable presheaves must come from an isomorphism of the representing objects :
corollary III: universality of representing objects
A presheaf is representable precisely if the comma category has a terminal object. If a terminal object is then .
This follows from unwrapping the definition of morphisms in the comma category and applying the Yoneda lemma to find
Hence says precisely that is a bijection.
For emphasis, here is the interpretation of these three corollaries in words:
corollary I says that the interpretation of presheaves on as generalized objects probeable by objects of is consistent: the probes of by are indeed the maps of generalized objects from into ;
corollary II says that probes by objects of are sufficient to distinguish objects of : two objects of are the same if they have the same probes by other objects of .
corollary III characterizes representable functors by a universal property and is hence the bridge between the notion of representable functor and universal constructions.
The Yoneda lemma tends to carry over to all important generalizations of the context of categories:
Necessity of naturality
The assumption of naturality is necessary for the Yoneda lemma to hold. A simple counter-example is given by a category with two objects and , in which , the set of integers greater than or equal to , in which , the set of integers greater than or equal to , and in which composition is addition. Here it is certainly the case that is isomorphic to for any choice of , but and are not isomorphic (composition with any arrow is greater than or equal to , so cannot have an inverse, since is the identity on and ).
A finite counter-example is given by the category with two objects and , in which , in which , and composition is multiplication modulo 2. Here, again, it is certainly the case that is isomorphic to for any choice of , but and are not isomorphic (composition with any arrow is , so cannot have an inverse, since is the identity on and ).
The term Yoneda lemma originated in an interview of Nobuo Yoneda by Saunders Mac Lane at Paris Gare du Nord:
In Categories for the Working Mathematician MacLane writes that this happened in 1954.
Reviews and expositions include
A discussion of the Yoneda lemma from the point of view of universal algebra is in
- Vaughan Pratt, The Yoneda lemma without category theory: algebra and applications (pdf).