---
tag: 494YXJH790
kind: definition
name: Representable functor
msc: [18A25, 18A40]
defines: KCCHH87H6E
uses: [A2DE66937D, 28NQ0KRHD3, VJXS8AK67E, KYCPCC57FT, 0BSJ7RQ9BW, 00E0YZCERH]
sources:
  - url: https://en.wikipedia.org/wiki/Representable_functor
---

## Definition

Let $\mathcal{C}$ be a locally small [[category|A2DE66937D]] and let $\mathbf{Set}$ be the category of [[sets|0BSJ7RQ9BW]].
A [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathbf{Set}$ is *representable* if there are an object $A$ of $\mathcal{C}$ and a natural isomorphism $\Hom(A, -) \cong F$.
A natural isomorphism is a [[natural transformation|KYCPCC57FT]] whose components are all bijections.
A functor $G \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ is *representable* if there are an object $A$ and a natural isomorphism $\Hom(-, A) \cong G$.
Here $\Hom(A, -)$ and $\Hom(-, A)$ are the [[hom functors|VJXS8AK67E]], and $A$ is called a *representing object*.

## Remarks

A representation $\Phi \colon \Hom(A, -) \cong F$ is determined by the element $u = \Phi_A(\id_A) \in F(A)$, called a universal element.
The pair $(A, u)$ represents $F$ exactly when for every object $X$ and every $x \in F(X)$ there is a unique $f \colon A \to X$ with $F(f)(u) = x$.

Example: the contravariant [[power set|00E0YZCERH]] functor $P \colon \mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$ sends $X$ to its set of subsets and $f \colon X \to Y$ to the preimage map $S \mapsto f^{-1}(S)$.
It is represented by $A = \{0, 1\}$: a function $\chi \colon X \to \{0, 1\}$ corresponds to the subset $\chi^{-1}(1)$, and this bijection commutes with preimages.

Non-example: the constant functor $F \colon \mathbf{Set} \to \mathbf{Set}$ with value the empty set is not representable.
For every set $A$ the set $\Hom(A, A)$ contains $\id_A$, so it is not in bijection with $F(A) = \emptyset$.