Difference between revisions of "Yoneda Lemma"
(statement and proof) |
m (fix latex typo) |
||
Line 24: | Line 24: | ||
& F(A) \\ | & F(A) \\ | ||
\downarrow h_A f && \downarrow Ff \\ | \downarrow h_A f && \downarrow Ff \\ | ||
− | h_A(B) & \! \stackrel{\phi_B}{\longrightarrow}\! & F(B) \end{ | + | h_A(B) & \! \stackrel{\phi_B}{\longrightarrow}\! & F(B) \end{array} </cmath> |
commutes for any arrow <math>f : A \to B</math> in <math>\mathcal{C}</math>. | commutes for any arrow <math>f : A \to B</math> in <math>\mathcal{C}</math>. | ||
In particular, we have | In particular, we have |
Latest revision as of 17:21, 6 November 2017
The Yoneda lemma is a result in category theory.
Contents
Statement
Let be a locally small category, and let be a functor from to Set, the category of sets. Let denote the functor that sends every object to and that takes the arrow to the function given by . In other words, is the hom functor . Then there exists a bijection between the set of natural transformations from to and the set . In symbols,
Proof
Let be a natural transformation. Then for each object of , gives us an arrow in the category Set, i.e., a function , such that the diagram commutes for any arrow in . In particular, we have But is the map from to . We thus have Thus for every object of , the morphism is uniquely determined by the element . Thus the map is an injection from to .
It thus remains to be shown that for any , the maps for every object of define a natural transformation. But this is true, as for any objects and of , any morphism of , and any element of , we have since is a functor. Thus the diagram commutes, so is a natural transformation, as desired.
Therefore the map is a bijection between and , as desired.
Dual Statement
If we replace the category with its opposite category , then we get the following result: if is a contravariant functor from to Set, then for every object of . This dual statement is also sometimes known as the Yoneda lemma.