The Yoneda Lemma

Math Online Tom Circle

Representable Functor F of C ( a, -):

$latex boxed {(-)^{a} = text {F} iff a = text {log F}}&fg=aa0000&s=3$

4.2Yoneda Lemma

Prove :

Yoneda Lemma:
$latex text {F :: C} to text {Set}$

$latex boxed {alpha text { :: [C, Set] (C (a, -),F) } simeq text {F a}}&fg=0000aa&s=3$

$latex alpha : text {Natural Transformation}$
$latex simeq : text {(Natural) Isomorphism}$

Proof: By “Diagram chasing” below, shows that

Left-side
: $latex alpha text { :: [C, Set] (C (a, -),F) } $ is indeed a (co-variant) Functor.

Right-side: Functor “F a“.

Note: When talking about the natural transformations, always mention their component “x”: $latex alpha_{x}, beta_{x}$

Yoneda Embedding (Lagatta)

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