Yoneda lemma

The lemma by Yoneda, by Nobuo Yoneda, is a mathematical statement from the branch of category theory. It describes the set of natural transformations between a Hom - functor and another functor.

The Yoneda lemma it, to transfer concepts that are familiar from the category of sets to arbitrary categories allowed.

Motivation

There were a category, set the category of sets and a functor. For each object category has the partial Hom - functor defined on objects and morphisms as follows:

• , One commonly used in this context is alternative notation for.

• .

If you have additional structures on the morphism ( enriched categories), such as in the case of abelian categories, so people like to replace the target category set of the Hom - functor by selecting the appropriate category, such as the category Ab of abelian groups. To then come back to the situation considered here, one has only behind the switch forgetful.

One can now ask the question, what natural transformations between the functors and on the set made ​​. Here the following Yoneda lemma gives an answer:

Statement

If a functor and an object, a bijection from the set of all natural transformations is in the crowd.

To this end, note that a natural transformation definition assigns to each object a morphism with certain compatibility conditions are satisfied (see natural transformation ). In particular, one has a morphism in the category Set ( simply called an illustration), so you can actually form as in the above lemma and receive an item. Hence, the map is well defined; they are also called the Yoneda map or the Yoneda isomorphism.

The proof is simple and illuminates the situation in the Yoneda lemma; Therefore, it is reproduced here: Is from a natural transformation, object, and that is is a morphism, then the following diagram is commutative by definition of natural transformation:

As a result.

Therefore, it is already uniquely determined by and, hence the injectivity of the Yoneda image. This formula is also used to surjectivity. Namely, we define for each object from the picture by. Then one can verify that this is a natural transformation is defined from to, which is mapped to under the Yoneda image. Pictures of the above mentioned type lead to the notion of representability of functors.

Yoneda embedding

As a simple application of the Yoneda lemma here the Yoneda embedding is treated. Yoneda the encapsulant used in the definition of the objects and Ind Pro objects.

Is a category, we denote the category of functors with natural transformations as morphisms. Note to the fact that the natural transformations between two functors and after the Yoneda lemma form a lot, it is therefore actually a Category. Next is denoted by the dual category. In this situation, we define the functor by the following data:

• , The functors are the objects in the.
• For a morphism is defined by where. Then a natural transformation, ie a morphism in.

Easily checked one after that this will actually defines a functor. The dual category is selected on the left, otherwise would run "in the wrong direction." It is now

• Yoneda embedding: The functor is fully faithful embedding.

Swaps are the roles of and, we obtain a fully faithful embedding.

The proof consists in an application of the Yoneda lemma. For full loyalty must be demonstrated that the pictures

Are bijective. For, that is for a natural transformation, that is, the mapping defines a mapping Yoneda

Since these illustration from Yoneda 's lemma is bijective, and because of all the following applies:

And is therefore also bijective. Therefore is fully faithful.

In order to see that even is an embedding, the injectivity of the functor on the category of objects must be shown (see Article faithful functor ). Are and two different objects, so true because a morphism can not have two different domains, and it follows, that is. Therefore, it is also an embedding.