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The Yoneda lemma suggests that instead of studying the (small) category C, one should study the category of all functors of C into Set (the category of sets with functions as morphisms). Set is the category we understand best, and a functor of C into Set can be seen as a "representation" of C in terms of known structures. The original category C is contained in this functor category, but new objects appear in the functor category which were absent and "hidden" in C. Treating these new objects just like the old ones often unifies and simplifies the theory.
This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modulesAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of over that ring. The ring takes the place of the category C, and the category of modules over the ring is a category of functors defined on C.
We denote by Fun(Cop,Set) the category of contravariant functorsCategory theory In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. Functor categories are of interest for two main re from C to Set (Here Cop is the opposite category of C). The morphisms in this category are natural transformations; we will write Nat(F,G) for the set of all natural transformations from the functor F to the functor G.
If A is an object of C, then we can assign to every object X of C the set of morphisms Mor(X,A). Every morphism φ : X → Y in C induces a map Mor(Y,A) → Mor(X,A) by the rule f |→ fφ. We have thus defined a contravariant functor Mor( - ,A) from C to Set, i.e., an element of Fun(Cop,Set). Such a functor is called a representable functor for C; often denoted hA.
The assignment
yields a covariant functor
This functor is called the Yoneda embedding and it is "natural" in the sense that every functor C → D induces a commutative diagramHomological algebra Category theory In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtai
of the corresponding Yoneda embeddings.
The content of the Yoneda lemma is that Y is indeed a full embedding, i.e., for all objects A, B in C, the functor Y induces a bijectionIn mathematics, a bijection bijective function or one-to-one correspondence is a function that is both injective ("one-to-one") and surjective ("onto"), and therefore bijections are also called one-to-one and onto . Intuitively, a bijective function creat
In other words: the morphisms between A and B in the original category C are "the same" as the ones between the two corresponding objects Y(A), Y(B) in the extended category Fun(Cop,Set).
And even more: for any contravariant functor F : C → Set and for any object A in C, there is a natural bijection
which means that, if you know how the functor F behaves on C, then you also know how it relates to the image of C in the extended category.
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