Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 25 / 3 / ii / Solution

For each $d\in \mathcal{D}$, let $P(d)={\pi }_0(d\downarrow F)$, the collection of connected components, where connectedness means connection by a zigzag of arrows. Precomposition with $u:d\to d^{\prime }$ gives a function $P(u):P(d^{\prime })\to P(d)$. Identity and composition are inherited from precomposition.

Construct a category $\mathcal{E}$ whose objects are pairs $(d,c)$ with $c\in P(d)$. An arrow $(d,c)\to (d^{\prime },c^{\prime })$ is an arrow $u:d\to d^{\prime }$ satisfying $c=P(u)c^{\prime }$. Composition is that of $\mathcal{D}$, and the component equation is preserved by composition. The projection $p:\mathcal{E}\to \mathcal{D}$ is a discrete fibration: an arrow $u:d\to d^{\prime }$ has precisely one lift into $(d^{\prime },c^{\prime })$, namely the arrow from $(d,P(u)c^{\prime })$ represented by $u$.

Define $J:\mathcal{C}\to \mathcal{E}$ byThis is well defined because $(a,1_{Fa})$ and $(a^{\prime },Fj)$ lie in the same component of $(Fa\downarrow F)$ whenever $j:a\to a^{\prime }$. Clearly $pJ=F$.

For an object $(d,c)$ of $\mathcal{E}$, an object of $((d,c)\downarrow J)$ is exactly a pair $(a,u:d\to Fa)$ representing the component $c$. Its morphisms are exactly the arrows in $(d\downarrow F)$ between such objects. Thus this comma category is the component $c$ itself, which is nonempty and connected. This proves that $J$ is a final functor, and henceThis is the final-discrete-fibration factorization. For small categories the collections $P(d)$ are sets and this construction is the category of elements of the presheaf $P$. For categories of a larger size, the construction is interpreted in the same chosen ambient size framework.