representable functor

on cateogry $\mathbf{Set}$
- representable functors $y^{S}: \mathbf{Set} \to \mathbf{Set}$ $y^{S} : Set \to Set$ , where $S$ $S$ is a set
  - for each set $X$ , it sends it to the homset of morphisms from $S$ to $X$ , $X^{S}:= \mathbf{Set}(S, X)$
  - for a function $h: X \to Y$ , it sends it to $h^{S}: X^{S} \to Y^{S}$
  - we say $S$ is the representing set of $y^{S}$
  - $y$ stands for Yoneda; we also call $y^{S}$ pure powers
  - representable functors are working on any category $\mathcal{C} \to \mathbf{Set}$ $C \to Set$ , here we only care about $\mathcal{C}$ $C$ being $\mathbf{Set}$ $Set$
    - more general definition in Category Theory for the Sciences:
      - a functor represented by $c \in \mathcal{C}$ , we say the functor $Y_{c}$ sends every object $d \in \mathcal{C}$ to $\text{Hom}_{\mathcal{C}}(c, -)$ . It acts similarily on morphisms $d \to d'$
    - so, given a category $\mathcal{C}$ and an object $c \in \text{Ob}(\mathcal{C})$ , we get a representable functor $Y_{\mathcal{C}}$
maps between representable functors are natural transformations
- Proposition: for any function $f: R \to S$ $f : R \to S$ , there is an induced natural transformation $y^{f}: y^{S} \to y^{R}$ $y^{f} : y^{S} \to y^{R}$ , where on any set $X$ $X$ , its $X$ $X$ -component … is giving by …
  - proof: shows the naturality square commutes
  - :
    - R and S are set
    - $y^{S}$ is the representable functor of S
    - $y^{R}$ is the representable functor of R
- the representable functor and natural transformation lives in a larger category $\mathbf{Set}^{\mathbf{Set}}$
$\mathbf{Set}^{\mathbf{Set}}$
- objects: functors $\mathbf{Set} \to \mathbf{Set}$
- morphisms: natural transformation between them

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