Yoneda embedding
The Yoneda embedding
- maps an object
π β π’ γ π = π’ ( β , π ) - maps a morphism
π : π β π γ π = π β π
Proof of embedding
Let
. By the Yoneda lemma we have π , π β π’ H π , γ π : { π’ π¨ π© } ( γ π , γ π ) β ( γ π ) π = π’ ( π , π ) where using the notation of the ^proof given
and π₯ β ( γ π ) π = π’ ( π , π ) π β π’ ( π β² , π ) ( π π₯ ) π β² π = ( ( γ π ) π ) π₯ = π’ ( π , π ) π₯ = π₯ π = π’ ( π β² , π₯ ) π = ( γ π₯ ) π β² π so
, implying π π₯ = γ π₯ H β 1 π , γ π = γ βΎ π’ ( π , π ) so in particular,
is fully faithful. γ is also clearly injective on objects, since if γ then γ π = γ π 1 π β π’ ( π , π ) = ( γ π ) π = ( γ π ) π = π’ ( π , π ) so
. π = π