Franklin's Notes


Representable functor

A covariant functor $F:\mathsf{C}\to\mathsf{Set}$, where $\mathsf{C}$ is a locally small category $\mathsf{C}$, is called representable if there exists an object $c\in\mathsf{C}$ such that $F$ is naturally isomorphic to the Hom-functor $\mathsf{C}(c,-)$. If $F$ is contravariant, we require instead a natural isomorphism between $F$ and $\mathsf{C}(-,c)$. In either case, we say that $F$ is represented by $c$, and the object $c$ together with the necessary natural isomorphism constitute a representation for $F$.

Here are some examples stolen from Riehl, with a bit of extra elaboration:

1. The forgetful functor $U:\mathsf{Group}\to\mathsf{Set}$ is represented by $\mathbb{Z}$, or the free group on a single generator. The natural transformation comprising a representation for $U$ has components $\alpha_G:UG\to\mathsf{Group}(\mathbb{Z},G)$ sending $g\in UG$ to the unique group homomorphism $f_g\in \mathsf{Group}(\mathbb{Z},G)$ with $f_g(1)=g$.

- How do we know $\alpha$ is natural? If $h:G\to H$ is a group homomorphism, then for any $g\in G$, meaning that $\alpha_H(Uh)=\mathsf{Set}(\mathbb{Z},h)\alpha_G$.

- This is a bijection, and hence an isomorphism in $\mathsf{Set}$, because every group homomorphism with domain $\mathsf{Z}$ is determined uniquely by the image of $1\in\mathbb Z$.

2. The functor $U:\mathsf{Ring}\to\mathsf{Set}$ is represented by $\mathbb{Z}[x]$, or the free unital ring on a single generator. The natural transformation comprising a representation for $U$ has components $\alpha_R:UR\to\mathsf{Ring}(\mathbb{Z}[x],R)$ sending $a\in UR$ to the unique ring homomorphism $f_a\in \mathsf{Ring}(\mathbb{Z}[x],R)$ with $f_a(x)=a$.

- How do we know $\alpha$ is natural? If $h:R\to S$ is a ring homomorphism, then for all $a\in R$, so we have that $\alpha_S(Uh)=\mathsf{Set}(\mathbb Z[x],h)\alpha_R$.

- This is a bijection because every ring homomorphism with domain $\mathsf{Z}[x]$ is determined uniquely by the image of $x$, since $x$ generates the ring $\mathsf{Z}[x]$.

3. The functor $(\cdot)^\times:\mathsf{Ring}\to\mathsf{Set}$ sending each ring to its set of units is represented by $\mathbb{Z}[x,x^{-1}]$, the ring of Laurent polynomials in one variable. The natural transformation comprising a representation for $(\cdot)^\times$ has components $\alpha_R:R^{\times}\to\mathsf{Ring}(\mathbb Z[x,x^{-1}],R)$ sending each unit $u\in R^\times$ to the unique map $f_u:\mathbb Z[x,x^{-1}]\to R$ sending $x\mapsto u$.

4. The functor $\text{Edge}:\mathsf{Dgr}\to\mathsf{Set}$ sending each simple digraph to its edge set, is represented by the digraph $D$ with two vertices and one edge. The natural transformation comprising a representation for $\text{Edge}$ has components $\alpha_G:\text{Edge}(G)\to\mathsf{SimpGph}(D,G)$ sending the edge $e\in \text{Edge}(G)$ to the unique graph homomorphism $f:D\to G$ mapping the unique edge of $D$ onto $e$.

5. Let $\text{Part}:\mathsf{Dgr}^{\text{op}}\to\mathsf{Set}$ be the contravariant functor that sends each digraph $G$ to the set of partitions $(V_1,V_2)$ of its vertex set such that every edge of $G$ originates at a vertex in $V_1$ and terminates at a vertex in $V_2$ (essentially capturing all of the ways in which $G$ can be written as a unidirectional bipartite digraph). Then $\text{Part}$ is also represented by $D$, the digraph with two vertices and one edge. The natural transformation comprising a representation for $\text{Part}$ has components $\alpha_G:\text{Part}(G)\to\mathsf{Dgr}(G,D)$, sending each partition $(V_1,V_2)$ to the unique graph homomorphism sending all vertices in $V_1$ to the initial vertex in $D$ and all vertices in $V_2$ to the final vertex in $D$.

category-theory

functors

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