Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 17 / 3 / i

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- Solution i

Solution

The Yoneda embedding sends

A

to

h_{A} = C (-, A)

and

u : A \to B

to postcomposition

h_{u} : h_{A} \to h_{B}

. Apply the Yoneda lemma with

X = h_{B}

:

Nat (h_{A}, h_{B}) ≅ h_{B} (A) = C (A, B) .

The inverse sends

u

precisely to

h_{u}

, because its component at

C

f : C \to A

to

u f

. Hence the induced maps on all hom-sets are bijections. The Yoneda embedding is full and faithful.

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