Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 4

The Yoneda embedding sends $c$ to the representable presheaf $H_c=\mathcal{C}(-,c)$. On a morphism $f:c\to d$, it sends a map $u:a\to c$ to $fu:a\to d$, giving $H_f:H_c\to H_d$.

The Yoneda lemma states that for every categorical presheaf $P$ there is a bijection, natural in both variables,Its inverse sends $x\in P(c)$ to the transformation $u:a\to c\mapsto P(u)x$. Apply it to $P=H_d$ to obtainUnder this bijection $H_f$ corresponds exactly to $f$, so the map induced by the embedding on each hom-set is bijective. Therefore the Yoneda embedding is full and faithful.

The equations $pi=1_A$ give $(ip)i=i$, so $i$ equalizes $ip$ and $1_B$. If $h:X\to B$ satisfies $(ip)h=h$, put $k=ph:X\to A$. Then $ik=iph=h$. If $k^{\prime }$ is another such factorization, applying $p$ gives $k^{\prime }=pi{k}^{\prime }=ph=k$. This provesIn particular the inclusion in a retract in a category is a monomorphism.

By part (b), $i:P\to H_B$ is the equaliser of $ip$ and $1_{H_B}$. Full faithfulness of the Yoneda embedding writes $ip=H_b$ for a unique endomorphism $b:B\to B$. Choose an equaliser $e:A\to B$ of $b$ and $1_B$ in $\mathcal{C}$.

For each $c$, the maps $c\to A$ are precisely the maps $u:c\to B$ with $bu=u$. Thus $H_e:H_A\to H_B$ is the pointwise equaliser of $H_b$ and $1_{H_B}$, hence their equaliser in the presheaf category. Both $i$ and $H_e$ have the same universal property, so their vertices are isomorphic. This proves the representable retract criterion:

Use 0-presentable object in the specified sense: its hom-functor preserves all small colimits. For a representable $H_c$, the Yoneda lemma identifies this hom-functor with evaluation at $c$. Given any small diagram $P_j$, pointwise colimits giveThese are the canonical comparison bijections, including for an empty diagram. Therefore every representable presheaf is 0-presentable, or a small projective object in this sense.

By the canonical colimit presentation of a presheaf, writewith structure maps ${\alpha }_{c,x}:H_c\to P$. Since $P$ is a 0-presentable object, the canonical mapis bijective. Every element of a set-valued colimit has a representative in some component. In particular $1_P$ is the image of a transformation $s:P\to H_c$ for some $(c,x)$, soThus $P$ is a retract of $H_c$. The representable retract criterion from part (c), using the assumed equalisers in $\mathcal{C}$, now gives $P$ is representable. Only existence of the representative of $1_P$ is needed; no stronger uniqueness-of-factorization assertion about individual colimit injections has been assumed.