Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 18 / 3

A functor $U:\mathcal{C}\to \mathbf{Set}$ is a representable functor if some object $R$ admits a natural isomorphism $\mathcal{C}(R,-)\cong U$. Here the representable is covariant.

For the identity functor on the Category of sets, choose the singleton $1=\{\ast \}$. The evaluation mapsare bijections with inverse $x\mapsto (\ast \mapsto x)$. For $u:X\to Y$, evaluation of $uf$ is $u(f(\ast ))$, proving naturality. Hence the identity functor on sets is represented by a singleton.

For a set $S$, let ${\iota }_s:1\to S$ select $s$. Given any family $f_s:1\to X$, define $f:S\to X$ by $f(s)=f_s(\ast )$. Then $f{\iota }_s=f_s$, and these equations determine every value of $f$, so it is unique.

This is precisely the universal property of a coproduct in a category, givingThe empty set gives the empty coproduct in a category, namely the initial object of the Category of sets.

Suppose $F:\mathbf{Set}\to \mathcal{C}$ is a left adjoint to $U$. The adjunction gives bijections, natural in $X$,The second map is evaluation at the singleton element, as in part (a). Thus $U$ is represented by $F1$. This argument uses the one-point set as a generator of the particular set-valued adjunction; it does not claim that every arbitrary right adjoint is representable.

Let $U\cong \mathcal{C}(R,-)$ be a representable functor, and let $(L,p_j)$ be the categorical limit of a small diagram $D:J\to \mathcal{C}$. A morphism $R\to L$ is uniquely equivalent to a family $f_j:R\to D(j)$ satisfying $D(u)f_j=f_k$ for every $u:j\to k$.

Such compatible families are exactly the elements of the categorical limit of the set-valued diagram $\mathcal{C}(R,D(-))$. Consequently the canonical comparisonis a bijection. Transporting it through the representing natural isomorphism proves that $U$ preserves every small limit that exists in $\mathcal{C}$. This proves that covariant representables preserve limits. For an empty diagram this says that maps into a terminal object form a singleton.

Choose a representing object $R$ and a natural isomorphism ${\theta }_X:\mathcal{C}(R,X)\cong UX$. Using the assumed small coproducts in a category, defineFor a function $v:S\to S^{\prime }$, define $F(v)$ by $F(v){\iota }_s={\iota }_{v(s)}$. The coproduct in a category uniqueness clause proves preservation of identities and composition, so this is a functor.

Restriction to the coproduct summands, followed by $\theta$, givesThese bijections are natural in $S$ by the definition of $F(v)$, and natural in $X$ by naturality of $\theta$. They establish $F⊣U$, the left adjoint to a covariant representable functor. The empty set is sent to the empty coproduct.