Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 1

Write $H_A=\mathcal{C}(-,A)$ for a representable presheaf. The contravariant Yoneda lemma states that, for every presheaf on a category $X$, evaluation at the identity is a bijectionnatural in both $A$ and $X$. Smallness of $\mathcal{C}$ ensures the relevant families and natural-transformation collections are sets.

For $x\in X(A)$ define ${\alpha }_B^x(h)=X(h)(x)$ for each $h:B\to A$. This family is a natural transformation: for $u:B^{\prime }\to B$, $X(u){\alpha }_B^x(h)=X(u)X(h)x=X(hu)x={\alpha }_{B^{\prime }}^x(hu)$. Its identity value is $x$. Conversely, naturality of any $\alpha$ at $h:B\to A$ gives ${\alpha }_B(h)=X(h){\alpha }_A(1_A)$. Thus evaluation and the displayed construction are inverse.

For a map $v:A\to A^{\prime }$, precomposing a transformation $H_{A^{\prime }}\to X$ with postcomposition $H_A\to H_{A^{\prime }}$ corresponds to applying $X(v):X(A^{\prime })\to X(A)$. For a transformation $\tau :X\to Y$, postcomposition corresponds to $x\mapsto {\tau }_A(x)$. These identities prove the stated naturality in both variables.

The Yoneda embedding is the functor $H_{\bullet }:\mathcal{C}\to [{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ sending $A$ to $H_A=\mathcal{C}(-,A)$ and $f:A\to B$ to postcomposition, $(H_f{)}_U(h)=fh$. Composition and identities are preserved by associativity and the identity laws in $\mathcal{C}$.

Apply the Yoneda lemma with $X=H_B$. It gives a bijection $\mathrm{Nat}(H_A,H_B)\cong \mathcal{C}(A,B)$. The inverse sends $f$ precisely to $H_f$, since $H_B(h)(f)=fh$. Hence every transformation between the representables comes from exactly one morphism of $\mathcal{C}$. This proves that $H_{\bullet }$ is a full and faithful functor.

If $f:A\to B$ is a monomorphism, every component $\mathcal{C}(U,A)\to \mathcal{C}(U,B)$ of $H_f$ is injective by left cancellation. Thus if $H_f\alpha =H_f\beta$ for two natural transformations into $H_A$, componentwise injectivity forces $\alpha =\beta$, so $H_f$ is monic.

Conversely suppose $H_f$ is monic, and let $a,b:U\to A$ satisfy $fa=fb$. Functoriality gives $H_f{H}_a=H_f{H}_b$, hence $H_a=H_b$. The Yoneda embedding is faithful, so $a=b$. This proves

If $f:A\to B$ has a section $g:B\to A$, functoriality gives $H_f{H}_g=1_{H_B}$. Thus $H_f$ is a split epimorphism and in particular an epimorphism.

For the converse, justify the needed pointwise surjectivity directly. Let $I\subseteq H_B$ be the image subpresheaf of $H_f$. Form a new presheaf by taking, at every object, two copies of $H_B(U)$ and identifying their elements in $I(U)$. Restriction maps descend because $I$ is a subpresheaf. The two canonical transformations $j_1,j_2:H_B\to H_B{⨿}_I{H}_B$ agree after $H_f$. If any component of $I$ is proper, an element outside it distinguishes $j_1$ from $j_2$. Therefore epimorphicity forces every component of $H_f$ to be surjective.

In particular, its component at $B$ has $1_B$ in its image: there is $g:B\to A$ with $fg=1_B$. This proves the Yoneda embedding detects split epimorphisms criterionOrdinary epimorphicity of $f$ alone would not suffice.