Hom-Functors and Representables

A category records objects by the maps between them, so the natural way to study one object is to collect every map into or out of it. Fix an object $A$ of a category $\mathcal{A}$. For each object $B$ there is a set $\mathcal{A}(A,B)$ of maps $A\to B$, and this assignment is functorial in $B$: a map $g:B\to B^{\prime }$ turns a map $A\to B$ into a map $A\to B^{\prime }$ by composition. The resulting functor $\mathcal{A}(A,-):\mathcal{A}\to \mathbf{Set}$ records how $A$ maps into the rest of the category.¹

Different objects give different records, and an object whose record matches a given functor $X$ is said to represent $X$. Representable functors are the second route to universal properties, after adjunctions: a representation of $X$ is a single object that captures everything $X$ does. The Yoneda lemma computes exactly how much information a hom-functor carries.

The covariant hom-functor

Fix a locally small category $\mathcal{A}$ and an object $A\in \mathcal{A}$, so that each class $\mathcal{A}(A,B)$ is a genuine set. Local smallness is what makes the following construction land in $\mathbf{Set}$.

The value $H^A(g)$ is post-composition with $g$; it is also written $g\circ -$ or $g_{\ast }$. Functoriality is immediate: $H^A(1_B)$ post-composes with an identity and so is the identity function, and $H^A(g^{\prime }\circ g)$ post-composes with $g^{\prime }\circ g$, which by associativity is post-composing with $g$ then with $g^{\prime }$, i.e. $H^A(g^{\prime })\circ H^A(g)$.

The name $H^A$ is fixed but many books write it $\mathcal{A}(A,-)$, $h^A$, or $\mathrm{Hom}(A,-)$; all denote the same functor.

The contravariant hom-functor

Fixing the second argument instead of the first gives the dual construction. Now a map $g:B^{\prime }\to B$ turns a map $B\to A$ into a map $B^{\prime }\to A$ by pre-composition, and pre-composition reverses direction, so the functor is contravariant.

The value $H_A(g)$ is also written $g^{\ast }$ or $-\circ g$. The reversal is essential: a map $B^{\prime }\to B$ downstairs induces a map $H_A(B)\to H_A(B^{\prime })$ upstairs, going the other way, exactly as a linear map of vector spaces induces a map of dual spaces in the reverse direction.

A functor ${\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ is called a presheaf on $\mathcal{A}$; the contravariant hom-functors $H_A$ are the representable presheaves, and they are the objects the Yoneda lemma is built around. The two variances are summarized below.

Representability

Only set-valued functors can be compared to a hom-functor, and the comparison is natural isomorphism.

A representation reduces a functor to a single representing object. Most functors into $\mathbf{Set}$ are not representable; the ones that are tend to be the functors that pick out structure by a single generator.

Standard representables. The recurring examples share a pattern: the representing object is a free structure on one generator.²

• Identity on $\mathbf{Set}$. A map $1\to B$ from the one-point set is an element of $B$, so $H^1(B)=\mathbf{Set}(1,B)\cong B$ naturally. The identity functor $1_{\mathbf{Set}}$ is represented by $1$.
• Forgetful $\mathbf{Top}\to \mathbf{Set}$. A continuous map from the one-point space is a point, so the underlying-set functor is $\mathbf{Top}(1,-)=H^1$.
• Forgetful $\mathbf{Grp}\to \mathbf{Set}$. For each $g\in G$ there is a unique homomorphism $\mathbb{Z}\to G$ sending $1\mapsto g$, so the underlying-set functor is $\mathbf{Grp}(\mathbb{Z},-)=H^{\mathbb{Z}}$, since $\mathbb{Z}$ is the free group on one generator.
• Forgetful ${\mathbf{Vect}}_k\to \mathbf{Set}$. A linear map $k\to V$ is determined by the image of $1\in k$, any vector at all, so ${\mathbf{Vect}}_k(k,V)\cong U(V)$ and $U=H^k$.
• Powerset $\mathcal{P}:{\mathbf{Set}}^{\mathrm{op}}\to \mathbf{Set}$. A subset of $B$ is a map $B\to 2$ into the two-element set, so $\mathcal{P}=H_2$ as a presheaf, with $\mathcal{P}(g)(U)=g^{-1}U$.
• Open sets $\mathcal{O}:{\mathbf{Top}}^{\mathrm{op}}\to \mathbf{Set}$. Continuous maps into the Sierpiński space $S$ (two points, one open singleton) correspond to open subsets, so $\mathcal{O}=H_S$.

Representables from adjunctions

Each forgetful functor above has a left adjoint (the free functor), and a left adjoint forces representability.

For $\mathbf{Top}$ the left adjoint is the discrete-space functor $D$ with $D(1)\cong 1$, recovering $U\cong H^1$; for ${\mathbf{Vect}}_k$ it is the free-vector-space functor with $F(1)=k$, recovering $U\cong H^k$; for $\mathbf{CRing}$ it is the polynomial-ring functor with $F(1)=\mathbb{Z}[x]$, so the forgetful functor is $H^{\mathbb{Z}[x]}$, matching the fact that ring maps $\mathbb{Z}[x]\to R$ are the same as elements of $R$.

Generalized elements

An object of an abstract category has no elements in any set-theoretic sense, but maps into it play the role of elements once a shape is chosen.

The term is only a synonym for map, but it reorganizes the covariant hom-functor: $H^S=\mathcal{A}(S,-)$ sends each object to its set of generalized elements of shape $S$, and functoriality says a map $A\to B$ transports $S$-shaped elements of $A$ to $S$-shaped elements of $B$.

In $\mathbf{Set}$ a shape-$1$ element is an ordinary element and a shape-$\mathbb{N}$ element is a sequence. In $\mathbf{Top}$ the shape-$1$ elements are points and the shape-$S^1$ elements are loops, so a continuous map $A\to B$ carries loops to loops. In algebra, a solution of $x^2+y^2=1$ in a ring $R$ is the same thing as a generalized element of shape $\mathbb{Z}[x,y]/(x^2+y^2-1)$, because a pair $(a,b)$ with $a^2+b^2=1$ is a ring map from that quotient into $R$. Fixing a shape and reading off its generalized elements recovers a functor's values; representability tests whether those values match a hom-functor.

Assembling the hom-functors

The families $\{H^A\}$ and $\{H_A\}$ are not independent: a map between objects induces a map between their hom-functors, so each family is itself functorial. A map $f:A^{\prime }\to A$ induces a natural transformation $H^f:H^A\to H^{A^{\prime }}$, whose $B$-component sends $p:A\to B$ to $p\circ f:A^{\prime }\to B$; note the reversal, $f:A^{\prime }\to A$ producing $H^A\to H^{A^{\prime }}$. Dually $f:A\to A^{\prime }$ induces $H_f:H_A\to H_{A^{\prime }}$ by $p\mapsto f\circ p$.

Packaging these gives four functors, dual in pairs.

The functor $H_{\bullet }:\mathcal{A}\to [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$, sending each object $A$ to its representable presheaf $H_A$, is the Yoneda embedding. Both packaged functors involve one contravariance that cannot be avoided. A single bifunctor unifies them:

acting on a pair $(f,g)$ by $p\mapsto g\circ p\circ f$. Contravariant in the first variable, covariant in the second, it carries the same data as $H^{\bullet }$ and $H_{\bullet }$ presented at once.

The reversal in the first argument is what makes $\mathrm{Hom}$ contravariant there: a map $A_0\to A$ pre-composes to send an element of $\mathcal{A}(A,B)$ to one of $\mathcal{A}(A_0,B)$. This bifunctor also explains the naturality clauses in the definition of adjunction: $F⊣G$ holds exactly when the functors $\mathcal{B}(F(-),-)$ and $\mathcal{A}(-,G(-))$ from ${\mathcal{A}}^{\mathrm{op}}\times \mathcal{B}$ to $\mathbf{Set}$ are naturally isomorphic.³

The Yoneda lemma computes the natural transformations out of a representable presheaf, and its consequences show that $H_{\bullet }$ embeds $\mathcal{A}$ faithfully into its presheaf category, so that an object is determined by the maps into it.