Covariant
Given a category and a covariant functor , is representable iff there is some object s.t. the following is a natural isomorpism
F\cong \mathcal{C}[u,-] $$$u$ is the *representing object* or *representation* of $F$. $u$ must be [[Terminal Object|terminal]] of such objects. ## Contravariant Given a [[category]] $\mathcal{C}$ and a [[Functor|contravariant functor]] $F:\mathcal{C}^{\mathrm{op}}\to\mathbf{Set}$, $F$ is *representable* iff there is some object $U:\mathcal{C}$ s.t. there is the following [[natural isomorphism]]F\cong \mathcal{C}[-,u]
$u$ is the *representing object* or *representation* of $F$. $u$ must be [[Initial Object|initial]] of such objects. ## Discussion Representation means that we can talk about functors as objects. [[Yoneda Lemma|Yoneda]] states that objects are determined entirely by the functors they represent. ## Examples ### [[Vect (Category)|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 (Category)|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 (Category)|Set]] Let $\mathcal{P}\ X$ be the set of [[Subset|subsets]] of $X$. Then $\mathcal{P}$ is contravariantly represented by 2. ## See also - [[Yoneda Lemma]] ## References - [[riehl2016-category-theory]]