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

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.