/sci/ - Science & Math

[Anonymous No.8494811]

Let [math]F:{\mathcal{C}^{op}} \to \operatorname{Set} [/math]. Show there is a bijection [math]\operatorname{Hom} \left( {{h_X},F} \right) \cong F\left( X \right)[/math].

[Anonymous No.8494817]

https://en.wikipedia.org/wiki/Yoneda_lemma#Proof

[Anonymous No.8494822]

>>8494811
what are any of the things in the OP

[Anonymous No.8494836]

>>8494822
C is a category
C^op is the opposite category (formally reverse all the arrows)

Set is the category of sets
F is a functor

h_X is the functor sending an object to the set of morphisms from that object into X

[Anonymous No.8494840]

>>8494822
F is a functor between the opposite category of C to the category of sets
i believe that's called a presheaf?

and [math] h_X(A) = \text{Hom}(A,X) [/math]
where Hom(A,B) is the set of mappings from A to B

the set of mappings between two functors is a little different, maps between functors are special kind of maps called natural transformations

what OP is saying is that the set of natural transformations between the contravariant hom functor on X and a functor F is isomorphic to the image of X under F

it's literally the yoneda lemma

[Anonymous No.8495013]

>>8494817
Thank you. Apparently my professor only taught the Yoneda embedding.