Covariant
Given a category and a functor , is representable iff there is some object s.t. for all ,
F(X)\cong \mathcal{C}[X,U] $$$U$ is the representing object of $F$. $U$ must be terminal of such objects. ## Contravariant Given a [[category]] $\mathcal{C}$ and a [[contravariant functor]] $F:\mathcal{C}^{\mathrm{op}}\to\mathbf{Set}$, $F$ is *representable* iff there is some object $U:\mathcal{C}$ s.t. for all $X:\mathcal{C}$,F(X)\cong \mathcal{C}[X,U]
$U$ is the representing object of $F$. $U$ must be initial of such objects. ## Discussion Representation means that we can talk about functors as objects. Yoneda states that objects are determined by the functors they represent. ## Examples ### Vect Given $V:\mathbf{Vect}_k$. Consider the functor $F:\mathbf{Vect}_k\to\mathbf{Set}$; $FX=\mathbf{Vect}_k[V,X]$ Then $F$ is represented by $V$. ### Top Let $FX$ be the set of [[connected components]] of $X$. $F:\mathbf{Top}\to \mathbf{Set}$ $FX=\pi_0(X)$. Then $\pi_0$ is representable by the discrete two point space $2=\{0,1\}$, because $\mathbf{Top}[2,X]\cong \pi_0(X)$ ### Set Let $P_f(X)$ be the set of finite subsets of $X$.