Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 20 / 1

For a categorical presheaf $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$ and $A\in \mathcal{C}$, the Yoneda lemma gives a natural bijectionTo construct its inverse, take $x\in X(A)$ and set ${\theta }_U^x(h)=X(h)x$ for $h:U\to A$. For $k:V\to U$, the functor law gives $X(k){\theta }_U^x(h)=X(hk)x={\theta }_V^x(hk)$, so ${\theta }^x$ is a natural transformation. Conversely, naturality of any $\theta$ at $h:U\to A$ gives ${\theta }_U(h)=X(h){\theta }_A(1_A)$. Thus the two constructions are inverse, proving the bijection.

The bijection is natural in $X$: a natural transformation $t:X\to Y$ sends the distinguished element $x$ to $t_A(x)$ on either side. It is natural in $A$: for $f:A\to B$, precomposing a transformation $\mathcal{C}(-,B)\to X$ with postcomposition by $f$ sends its distinguished element $x\in X(B)$ to $X(f)x$. These equations prove the claimed naturality, rather than just an objectwise correspondence. Reversing morphisms gives the covariant form $\mathrm{Nat}(\mathcal{C}(A,-),Z)\cong Z(A)$ for $Z:\mathcal{C}\to \mathbf{Set}$.

Define the Yoneda embedding $H_{\bullet }:\mathcal{C}\to [{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ by $H_A(U)=\mathcal{C}(U,A)$ and $H_A(k)(h)=hk$. For $f:A\to B$, define $H_{\bullet }(f{)}_U(h)=fh$. Associativity proves naturality in $U$, and preservation of identity morphisms and composition in a category proves that $H_{\bullet }$ is a functor.

Apply the Yoneda lemma to $X=H_B$. It givesThe natural transformation corresponding to $f$ has component $h\mapsto fh$, exactly $H_{\bullet }(f)$. Therefore the induced map on every hom-set is a bijection: the Yoneda embedding is full and faithful.

If $f:A\to B$ is a monomorphism, each function $\mathcal{C}(U,A)\to \mathcal{C}(U,B)$ given by postcomposition with $f$ is injective. Consequently, if two natural transformations into $H_A$ become equal after $H_{\bullet }(f)$, their components are equal, and the transformations themselves are equal. This proves that $H_{\bullet }(f)$ is a monomorphism in the presheaf category.

Conversely, suppose $H_{\bullet }(f)$ is a monomorphism and $u,v:U\to A$ satisfy $fu=fv$. Then $H_{\bullet }(f)H_{\bullet }(u)=H_{\bullet }(f)H_{\bullet }(v)$. Cancellation gives $H_{\bullet }(u)=H_{\bullet }(v)$, and faithfulness of the Yoneda embedding gives $u=v$. Hence $H_{\bullet }(f)$ is monic exactly when $f$ is monic.

We first justify the relevant pointwise epimorphism in a functor category criterion. A componentwise surjective function gives an epimorphism, because equality of two composites can be cancelled at each component. Conversely, for a natural transformation $t:X\to Y$, form the pointwise pushout $P=Y{⨿}_{X}Y$. Restrictions of $Y$ preserve the identifications imposed by $t$, so these pointwise sets and restrictions form a categorical presheaf. The two inclusions $j_1,j_2:Y\to P$ satisfy $j_{1}t=j_{2}t$. If $t$ is epic, $j_1=j_2$. At an object $U$, an element outside the image of $t_U$ would retain two distinct copies in $P(U)$, contradicting this equality. Thus every $t_U$ is surjective.

Apply this criterion to $H_{\bullet }(f)$ at $B$. Surjectivity of $\mathcal{C}(B,A)\to \mathcal{C}(B,B)$ supplies $g:B\to A$ with $fg=1_B$. Conversely, if $fg=1_B$, then $H_{\bullet }(f)H_{\bullet }(g)=1_{H_B}$, making $H_{\bullet }(f)$ a split epimorphism and hence an epimorphism. Therefore