LemmaProved

Yoneda lemma

Statement

Let $\mathcal{C}$ be a locally small category, $A$ an object of $\mathcal{C}$ , and $F:\mathcal{C}^{\mathrm{op}}\to\mathbf{Set}$ a functor. There is a bijection $\mathrm{Nat}(\mathrm{Hom}_{\mathcal C}(-,A),\,F)\cong F(A)$ , natural in both $A$ and $F$ . In particular, the Yoneda embedding $A\mapsto \mathrm{Hom}_{\mathcal C}(-,A)$ is fully faithful.

Why is it true?

An object is completely determined, up to isomorphism, by the web of arrows pointing into it from every other object in the category — you never need to look 'inside' an object, only at how it relates to everything else.

Proof sketch

Given a natural transformation $\eta:\mathrm{Hom}(-,A)\to F$ , evaluate it at the identity arrow $\mathrm{id}_A\in\mathrm{Hom}(A,A)$ to get an element $\eta_A(\mathrm{id}_A)\in F(A)$ . Conversely, given $u\in F(A)$ , define $\eta_X(f)=F(f)(u)$ for every arrow $f:X\to A$ . A direct check using naturality shows these two constructions are mutually inverse.

Topics that use this theorem

The Yoneda lemma Categories and functors

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Saunders Mac Lane (1978). Categories for the Working Mathematician
Nobuo Yoneda (1954). On the homology theory of modules