Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 18 / 2

The covariant form of the Yoneda lemma says that for a locally small category $\mathcal{C}$, a functor $F:\mathcal{C}\to \mathbf{Set}$ and an object $A$, there is a bijection, natural in $A$ and $F$,Explicitly its two directions areFor $u:B\to B^{\prime }$, functoriality gives $F(u){\alpha }_B^x(f)=F(uf)(x)={\alpha }_{B^{\prime }}^x(uf)$, so ${\alpha }^x$ is a natural transformation. Conversely, naturality of $\alpha$ at $f:A\to B$ gives ${\alpha }_B(f)=F(f)({\alpha }_A(1_A))$. This proves that the displayed maps are inverses.

Naturality in $F$ follows because postcomposition by $\theta :F\Rightarrow F^{\prime }$ sends $x$ to ${\theta }_A(x)$. For $u:A^{\prime }\to A$, precomposition of transformations by $\mathcal{C}(u,-)$ sends ${\alpha }^x$ to ${\alpha }^{F(u)(x)}$. This also verifies naturality in the representing object.

In the Category of sets, an epimorphism is exactly a surjective function. A surjective function is right-cancellable. If $q:X\to Y$ misses $y\in Y$, the constant-zero function and the function that is one at $y$ and zero elsewhere are distinct maps $Y\to \{0,1\}$ with equal composites with $q$.

If every ${\alpha }_A$ is surjective, equality $\beta \alpha =\gamma \alpha$ for natural transformations $\beta ,\gamma :G\Rightarrow H$ gives ${\beta }_A={\gamma }_A$ for every $A$. Thus $\alpha$ is an epimorphism in the functor category.

For the converse, construct the pointwise amalgamated doublewhere exactly the two copies of each element of $\mathrm{im}{\alpha }_A$ are identified; different elements of $G(A)$ remain different. Define $H(u)[y,i]=[G(u)y,i]$. Naturality of $\alpha$ implies that $G(u)$ takes its image into the image at the target, so this formula is well-defined and gives a functor. The maps $j_{i,A}(y)=[y,i]$ form natural transformations $j_0,j_1:G\Rightarrow H$, with $j_0\alpha =j_1\alpha$.

If $\alpha$ is an epimorphism, $j_0=j_1$. Since the two copies of an element outside $\mathrm{im}{\alpha }_A$ would be distinct, every element must lie in that image. Hence $\alpha$ is epic if and only if every component is epic. This proves the pointwise epimorphism in a functor category criterion directly, including the needed existence and naturality of the separating functor.

A projective object in a category is an object $P$ such that, for every epimorphism $e:X\to Y$ and every $u:P\to Y$, there is $v:P\to X$ with $ev=u$.

Let $P={\coprod }_{i\in I}{P}_i$ be a coproduct in a category of projective objects in a category, with injections ${\iota }_i$. Given $e:X\to Y$ epic and $u:P\to Y$, projectivity supplies $v_i:P_i\to X$ with $e{v}_i=u{\iota }_i$. Choose these lifts for the set-indexed family. The coproduct in a category supplies a unique $v:P\to X$ satisfying $v{\iota }_i=v_i$. Since $ev{\iota }_i=u{\iota }_i$ for every $i$, its universal property gives $ev=u$.

Thus coproducts of projective objects are projective. For an empty family, $P$ is the initial object, and the lifting assertion follows directly from its unique maps. The family-of-lifts step uses the usual axiom of choice.

Take an epimorphism $\alpha :F\Rightarrow G$ in $[\mathcal{C},\mathbf{Set}]$ and a natural transformation $\beta :\mathcal{C}(A,-)\Rightarrow G$. By the Yoneda lemma, $\beta$ corresponds to $x={\beta }_A(1_A)\in G(A)$. By the pointwise epimorphism in a functor category criterion, choose $y\in F(A)$ with ${\alpha }_A(y)=x$.

The Yoneda lemma gives ${\gamma }_B(f)=F(f)(y)$, a natural transformation $\mathcal{C}(A,-)\Rightarrow F$. Naturality of $\alpha$ yieldsTherefore covariant representables are projective in the set-valued functor category. Local smallness ensures that $\mathcal{C}(A,-)$ is set-valued.