The Yoneda Lemma
and are thus able to choose our functor LHS to guarantee that the naturality condition on α is satisfied. It remains only to show that LHS as defined here is indeed a functor. Showing that it preserves identities id A,F = (id A , id F ) is very simple: we have id A,F : ( A,F ) → ( A,F ), yielding
as required. Showing that LHS preserves composition is not much harder: for a morphism ( f, Φ) : ( X,F ) → ( Y,G ) and a morphism ( g, Ψ) : ( Y,G ) → ( Z,H ), we have
This completes the proof.
3 Consequences Having now seen the Lemma in all its glory, we are better equipped to see how it can be helpful. In this section I provide a glimpse of its power, alongside a very brief overview of how it can be further generalized. 3.1 Cayley’s Theorem The key result in group theory known as Cayley’s Theorem is a special case of the Yoneda Lemma. For a reader familiar with group theory, this section should show how category theory subsumes many other fields of study, while the definition of a group itself may also be of interest. For any other reader, this may serve as an example of how category theory effectively gives an understanding of other disciplines for free. Definition 10 A group is a category with only one object • , where every morphism is an isomorphism. That may seem like a simple definition, but even groups, a very special case of categories, are studied extensively. Intuitively, a group can represent some sort of symmetry, such as rotations of a triangle. A simple example of a group is the real numbers R, where each morphism is a number and composition is addition; with the previous intuition, this represents the symmetry of the number line under translation.
Definition 11 For any set n, the symmetric group S n is the group whose morphisms are exactly the isomorphisms on n in Set , with composition given as in Set .
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