Representable functor

A covariant functor is representable when it is naturally isomorphic to $\mathcal{C}(A,-)$ for some object $A$; a contravariant functor is representable when it is naturally isomorphic to $\mathcal{C}(-,A)$.

Suppose a small category has equalizers. If a categorical presheaf $P$ is a retract $P\stackrel{i}{\to }{H}_B\stackrel{p}{\to }P$, with $pi=1_P$, then $i$ equalizes $ip$ and $1_{H_B}$. The Yoneda lemma writes $ip=H_b$ for an endomorphism $b:B\to B$. An equalizer $A\to B$ of $b$ and $1_B$ gives the same presheaf equalizer, since the Yoneda embedding preserves categorical limits. Uniqueness of equalizers gives $P\cong H_A$.

On a complete category that is locally small, a set-valued functor is representable exactly when it preserves small categorical limits and its category of elements has a weakly initial set. Categorical limit preservation makes that comma category complete. The initial-object lemma for complete categories with a weakly initial set then supplies a universal element, hence a representation of a functor.

If $\mathcal{C}$ has small coproducts in a category and $U\cong \mathcal{C}(R,-)$, then $F(S)={\coprod }_{s\in S}R$ is a left adjoint to $U$. Maps from this coproduct to $X$ are families of maps $R\to X$, naturally equivalent to functions $S\to UX$. Conversely, a set-valued right adjoint $U$ is represented by $F(1)$, because maps from a singleton evaluate to elements.

A representable functor $\mathcal{C}(R,-)$ preserves every small categorical limit that exists. A map from $R$ into a limit is exactly a compatible family of maps from $R$ to the diagram objects, and such a family is exactly an element of the corresponding limit of sets. The comparison bijection is the one induced by the limit projections.

A categorical presheaf is representable if its elements projection has a colimit and the presheaf preserves the opposite-indexed categorical limit of that projection. Compatibility of all distinguished elements then supplies a universal element at the colimit object. The preservation requirement must cover this diagram even when its indexing category is large.

Let $\mathcal{C}$ be the ordered category of all ordinals with an added greatest object $\infty$. The categorical presheaf with $X(\alpha )=\{\ast \}$ at ordinals and $X(\infty )=\varnothing$ preserves every small categorical limit: a small supremum is an ordinal unless its diagram contains $\infty$. Its category of elements consists of all ordinals, and their projection has the large colimit $\infty$. Nevertheless $X$ is not representable: an ordinal representer has bounded support, while the representer $\infty$ has a nonempty value at $\infty$. Thus preservation must include the possibly large opposite elements diagram. The dual functor $X^{\mathrm{op}}:\mathcal{C}\to {\mathbf{Set}}^{\mathrm{op}}$ also disproves the comma-projection criterion for a right adjoint under small-colimit preservation alone.

Every functor $F:\mathcal{C}\to \mathbf{Set}$ on a small category is the colimit of the covariant representable functors $\mathcal{C}(A,-)$ indexed by the opposite of its category of elements. The cocone indexed by $(A,x)$ sends $f:A\to B$ to $F(f)(x)$. By the Yoneda lemma, a compatible cocone into $G$ is exactly a natural transformation $F\to G$. The opposite indexing direction is essential because precomposition reverses the representing-object arrow.

A representation of $F:\mathcal{C}\to \mathbf{Set}$ is an object $A$ together with a universal element $a\in F(A)$ such thatis a bijection for every $B$, naturally in $B$. Equivalently, it is a specified natural isomorphism $\mathcal{C}(A,-)\cong F$.

Two representations of a functor have unique mutually inverse morphisms carrying their specified universal elements to one another. Composing those morphisms preserves each universal element, so injectivity of the representing bijection makes the composites identities. Arbitrary isomorphisms of the underlying objects need not preserve the specified elements.

If $H:{\mathcal{A}}^{\mathrm{op}}\times \mathcal{B}\to \mathbf{Set}$ has chosen representations ${\psi }_B:\mathcal{A}(-,GB)\cong H(-,B)$, the Yoneda lemma determines a unique functor $G$ on morphisms by $\mathcal{A}(-,Gb)={\psi }_{B^{\prime }}^{-1}H(-,b){\psi }_B$. This makes the representing natural isomorphisms natural in $B$.

The universal element in a representation of a functor is the image of $1_A$ under the representing natural isomorphism. Any two universal elements induce unique mutually inverse maps between their representing objects, so representations are unique up to unique compatible isomorphism.

The Yoneda lemma gives natural bijectionsand their contravariant analogues.

For a presheaf on a category, the categorical coend identifies $(fh,x)$ with $(h,X(f)x)$. The displayed isomorphism sends the class of $(h:U\to W,x\in X(W))$ to $X(h)x$, with inverse $y\mapsto [1_U,y]$. The coend relation makes these inverse and natural. Consequently every presheaf is the colimit of representables indexed by its category of elements.

The Yoneda embedding sends $A$ to $\mathcal{C}(-,A)$. The Yoneda lemma identifies natural transformations $yA\to yB$ with morphisms $A\to B$, so the embedding is full and faithful. It preserves every limit that exists in $\mathcal{C}$.

For $f:A\to B$, the Yoneda embedding sends $f$ to postcomposition $\mathcal{C}(-,A)\to \mathcal{C}(-,B)$. If this is epic in the presheaf category, its component at $B$ is surjective, so $1_B=fg$ for some $g:B\to A$. Conversely such a section makes the natural transformation split epic. Pointwise surjectivity follows directly by doubling the target presheaf along the image subpresheaf: the two inclusions agree on the image but differ on any element outside it.

For a small Cartesian closed category, the Yoneda embedding into its presheaf category preserves exponential objects. At $C$, the exponential in a presheaf category has value $\mathrm{Nat}(YC\times YA,YB)$. Since $YC\times YA\cong Y(C\times A)$, the Yoneda lemma identifies this with $\mathcal{C}(C\times A,B)\cong \mathcal{C}(C,B^A)$. Every identification is natural in all three objects.

The covariant version of the Yoneda embedding sends an object to its outgoing representable functor. Precomposition reverses arrows. The Yoneda lemma gives $\mathrm{Nat}(Y(A),Y(B))\cong \mathcal{C}(B,A)$, so it is full and faithful. It carries existing colimits in $\mathcal{C}$ to pointwise categorical limits.