Representable Functors

Summary

A functor $F:\mathcal{A}\to \mathbf{Set}$ is representable if it is naturally isomorphic to the hom-functor $\mathcal{A}(A,-)$ for some object $A$. The representing object $A$ is unique up to isomorphism (by the Yoneda lemma), and the isomorphism $\mathcal{A}(A,-)\cong F$ corresponds to a distinguished “universal element” $u\in F(A)$.

Overview

The hom-functors $\mathcal{A}(A,-):\mathcal{A}\to \mathbf{Set}$ are the most natural functors associated to any category. A representable functor is one that “looks like a hom-functor,” i.e., one that is naturally isomorphic to some $\mathcal{A}(A,-)$. The theory of representable functors is the formal foundation for universal properties: to say that an object $A$ represents $F$ is to say that $A$ is the universal object for the construction $F$ describes.

Main Content

Definition 4.1.1: Representable Functor

Let $\mathcal{A}$ be a locally small category. A functor $F:\mathcal{A}\to \mathbf{Set}$ is representable if there exists $A\in \mathcal{A}$ and a natural isomorphism $\alpha :\mathcal{A}(A,-)\stackrel{\sim }{\to }F$.

The pair $(A,\alpha )$ is called a representation of $F$. The object $A$ is the representing object.

For contravariant functors $F:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, representability means $F\cong \mathcal{A}(-,A)$ for some $A$.

The Hom-Functors

Definition: Covariant Hom-Functor $H^A$

For $A\in \mathcal{A}$, the covariant hom-functor $H^A=\mathcal{A}(A,-):\mathcal{A}\to \mathbf{Set}$ sends:

• Object $B\mapsto \mathcal{A}(A,B)$
• Map $f:B\to C\mapsto f_{\ast }:\mathcal{A}(A,B)\to \mathcal{A}(A,C)$, where $f_{\ast }(g)=f\circ g$ (postcomposition)

Definition: Contravariant Hom-Functor $H_A$

For $A\in \mathcal{A}$, the contravariant hom-functor $H_A=\mathcal{A}(-,A):{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ sends:

• Object $B\mapsto \mathcal{A}(B,A)$
• Map $f:C\to B\mapsto f^{\ast }:\mathcal{A}(B,A)\to \mathcal{A}(C,A)$, where $f^{\ast }(g)=g\circ f$ (precomposition)

Definition: Bifunctor Hom

The hom bifunctor $\mathcal{A}(-,-):{\mathcal{A}}^{\mathrm{op}}\times \mathcal{A}\to \mathbf{Set}$ sends $(B,C)\mapsto \mathcal{A}(B,C)$ and $(f:B^{\prime }\to B,g:C\to C^{\prime })\mapsto (h\mapsto g\circ h\circ f)$.

Generalized Elements

Definition 4.1.25: Generalized Element

For $A,S\in \mathcal{A}$, a generalized element of $A$ of shape $S$ is a map $f:S\to A$. The set of all generalized elements of shape $S$ is $\mathcal{A}(S,A)=H^S(A)$.

For $S=1$ (terminal object, when it exists), generalized elements of shape $1$ are global elements (ordinary elements of $A$).

Generalized elements make the category-theoretic analogy with sets precise: in Set, maps $\{\ast \}\to X$ are exactly elements of $X$. In a general category, maps $S\to A$ are “elements of $A$ as seen from the perspective of $S$.”

Examples of Representable Functors

Example: Forgetful Functor on Groups (BCT, Ch. 4.1)

The forgetful functor $U:\mathbf{Grp}\to \mathbf{Set}$ is representable: $U\cong \mathbf{Grp}(\mathbb{Z},-)$. The bijection $U(G)\cong \mathbf{Grp}(\mathbb{Z},G)$ sends $g\in G$ to the unique homomorphism $\mathbb{Z}\to G$ with $1\mapsto g$.

Example: Underlying Set of a Ring (BCT, Ch. 4.1)

The forgetful functor $\mathbf{Ring}\to \mathbf{Set}$ is representable by the polynomial ring $\mathbb{Z}[x]$: $\mathbf{Ring}(\mathbb{Z}[x],R)\cong R$ as sets (a ring map $\mathbb{Z}[x]\to R$ is determined by where $x$ goes).

Example: Non-Representable Functor (BCT, Ch. 4.1)

The functor $F:\mathbf{Set}\to \mathbf{Set}$ sending $X\mapsto \mathcal{P}(X)/\text{finite}$ (power set modulo finite sets) is not representable, as it does not preserve filtered colimits.

Representation = Universal Element

Theorem: Representations via Universal Elements (BCT, Ch. 4.3, Cor. 4.3.2)

A representation of $F:\mathcal{A}\to \mathbf{Set}$ is equivalently a pair $(A,u)$ where $A\in \mathcal{A}$ and $u\in F(A)$ such that:

For every $B\in \mathcal{A}$ and every $x\in F(B)$, there is a unique $f:A\to B$ with $(Ff)(u)=x$.

The element $u\in F(A)$ is called the universal element of the representation.

The universal element corresponds to ${\alpha }_A(1_A)$ under the natural isomorphism $\alpha :\mathcal{A}(A,-)\cong F$. Every universal property is exactly a universal element of some functor.

Connections

• The Yoneda lemma (Yoneda Lemma) computes $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,X)\cong X(A)$ — natural transformations from a representable are determined by a single element.
• Limits (General Limits) are representable: $\lim D$ represents the functor $\mathrm{Cone}(-,D)$.
• Adjunctions (Adjoint Functors): $F⊣G$ iff $\mathcal{B}(F-,B)$ is representable by $GB$ for each $B$.

See Also

• Yoneda Lemma — The fundamental theorem of representability
• Yoneda Embedding and Consequences — Full faithfulness of the Yoneda embedding
• Adjoint Functors — Adjunctions as representability
• Limits via Representables — Limits as representable functors