Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 1

For $A\in \mathcal{C}$ and $F:\mathcal{C}\to \mathbf{Set}$, the Yoneda lemma gives a natural bijectionAssume $\mathcal{C}$ is small. For every pair $(A,x)$ with $x\in F(A)$, let ${\alpha }^{A,x}:\mathcal{C}(A,-)\to F$ be the corresponding natural transformation. Their copairing isAt an object $B$, the element $x\in F(B)$ is the image of $1_B$ in the summand indexed by $(B,x)$. The map is therefore pointwise surjective and hence an epimorphism in the functor category.

Let $P:\mathcal{E}\to \mathcal{D}$ be a discrete fibration and let $m:A\to A^{\prime }$ be monic in $\mathcal{E}$. If $u,v:B\to P(A)$ satisfy $P(m)u=P(m)v$, lift $u,v$ uniquely to $\tilde{u},\tilde{v}$ with codomain $A$. The composites $m\tilde{u},m\tilde{v}$ are lifts of the same arrow with codomain $A^{\prime }$, so uniqueness gives equality. Since $m$ is monic, $\tilde{u}=\tilde{v}$, hence $u=v$. Thus $P(m)$ is monic.

Use the convention that $(F\downarrow B)$ has objects $(A,u:FA\to B)$. Its forgetful functor sends $(A,u)$ to $A$. Given $h:C\to A$, the unique arrow above $h$ with codomain $(A,u)$ has domain $(C,uFh)$. Hence the forgetful functor is a discrete fibration.

Assume every morphism of $\mathcal{C}$ is monic. For a representable presheaf $F=\mathcal{C}(-,A)$, the category $(1\downarrow F)$ is the category of elements, equivalently the slice $\mathcal{C}/A$. A morphism from $f:B\to A$ to $g:C\to A$ is an $h:B\to C$ satisfying $gh=f$. Since $g$ is monic, there is at most one such $h$. Thus $(1\downarrow F)$ is a preorder.

Assume every category of elements of a representable presheaf is a preorder. LetA disjoint union of preorders is a preorder. The category-of-elements projection $P\to \mathcal{C}$ is a discrete fibration. It is surjective on objects because $B\in \mathcal{C}$ is the image of the object $1_B:B\to B$ in the summand indexed by $B$.

Suppose a preorder $P$ admits a discrete fibration $q:P\to \mathcal{C}$ that is surjective on objects. Given $f:B\to A$ in $\mathcal{C}$, choose $a\in P$ above $A$. The discrete-fibration property lifts $f$ to $\tilde{f}:b\to a$. Every arrow in a preorder is monic, and a discrete fibration preserves monomorphisms by part (b), so $f=q(\tilde{f})$ is monic. This proves the remaining implication and hence the equivalence.