Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 119 / 5

A Cartesian closed category has finite products in a category and, for every $F$, an exponential object $G^F$ characterized naturally byIn $[\mathcal{C},\mathbf{Set}]$, finite products and the terminal functor are computed pointwise. With $h_B=\mathcal{C}(B,-)$, defineThis is a set because $\mathcal{C}$ is a small category. For $v:B\to B^{\prime }$, precomposition $h_{B^{\prime }}\to h_B$ induces $(G^F)(v)$ by precomposition of natural transformations, making $G^F$ a covariant functor.

Given $\alpha :K\times F\to G$, define its curry $\hat{\alpha }:K\to G^F$ byFor $a:C\to C^{\prime }$, naturality of $\alpha$ gives $G(a){\alpha }_C(K(u)y,x)={\alpha }_{C^{\prime }}(K(au)y,F(a)x)$, proving that this is a natural transformation $h_B\times F\to G$. The same formula with $u$ precomposed proves naturality of $\hat{\alpha }$ in $B$.

Conversely, from $\beta :K\to G^F$ defineNaturality of $\beta$ and its values proves naturality of $\alpha$. Currying this result recovers $\beta$, since ${\beta }_C(K(u)y)=(G^F)(u)({\beta }_B(y))$; uncurrying a curry recovers $\alpha$ by setting $u=1_B$. These mutually inverse constructions are natural in $K,G$, giving the adjunction required for an exponential object. Therefore

For a tiny object $A$, the exponential functor $E_A=(-{)}^A$ has a right adjoint, say $R_A$. The defining exponential adjunction gives the natural isomorphismsince $(X^B{)}^A\cong X^{A\times B}$. If $A,B$ are tiny objects, the composite on the right is a left adjoint with right adjoint $R_B{R}_A$. Thus $A\times B$ is tiny. The terminal object is tiny because exponentiation by it is naturally the identity, which is its own right adjoint. Induction, including this empty-product case, proves tiny objects are closed under finite products:

Fix $A$, and let $S(B)=B⨿A$ be the functor obtained from the chosen binary coproduct in a category construction. Its arrow map is $S(v)=v⨿1_A$. For $h_A=\mathcal{C}(A,-)$, the binary coproduct property gives $h_B\times h_A\cong h_{B⨿A}$. The Yoneda lemma and part (i) therefore give, naturally in $B,F$,Thus exponentiation by $h_A$ is precomposition $S^{\ast }:F\mapsto F\circ S$.

This precomposition has a right adjoint given by Right Kan extension. Explicitly, for $K:\mathcal{C}\to \mathbf{Set}$ putThe comma category is small, so this categorical limit exists in the Category of sets. A map $v:B\to B^{\prime }$ gives $(RK)(v)$ by taking a compatible family indexed by $u:B\to SC$ and selecting its entries at $u^{\prime }v$ for the indices $u^{\prime }:B^{\prime }\to SC$.

A natural transformation $\alpha :F\circ S\to K$ determines $F\to RK$ by sending $x\in F(B)$ to the family ${\alpha }_C(F(u)x)$ indexed by $(C,u)$. Conversely, a map $F\to RK$ determines $F(SC)\to K(C)$ by projecting to the index $(C,1_{SC})$. Naturality and the limit's compatibility relations make these constructions inverse. Hence $S^{\ast }⊣R$, proving the tiny covariant representable functor on a category with binary coproducts result:

Let $P$ be a tiny object of $[\mathcal{C},\mathbf{Set}]$, and let $0$ be the initial object of $\mathcal{C}$. The representable functor $h_0=\mathcal{C}(0,-)$ is the terminal functor, since each of its values is a singleton. The exponential adjunction and the Yoneda lemma giveSince $P$ is tiny, $E_P:F\mapsto F^P$ is itself a left adjoint, so preserves all colimits. Evaluation at $0$ also preserves colimits, because they are pointwise in a set-valued functor category. Therefore $\mathrm{Nat}(P,-)$ preserves the coproduct construction.

It also preserves epimorphisms. Indeed, every such epimorphism is a pointwise epimorphism in a functor category, hence the coequalizer of its kernel pair, as can be checked pointwise in sets. A left adjoint preserves that coequalizer and therefore sends it to a regular epimorphism, which is an epimorphism; evaluation at zero preserves surjectivity. Thus $P$ is an irreducible projective in a set-valued functor category.

A Cauchy-complete category is one in which every idempotent morphism splits. By the characterization allowed in the question, irreducible projectives are representable functors for such an indexing category. Applying it to $P$ proves the tiny covariant functor on a Cauchy-complete category with an initial object is representable conclusion: