# Are final functors stable under pullback?

Recall the notion of a final functor, which is a sort of colimit-preservation property.
Is such class of functors stable under pullbacks in Cat? Namely, is the pullback of a final functor along any other functor still final? If not, what is a counterexample?

A reference would also be welcome.

Mathematics Asked by geodude on November 12, 2021

No, final functors are not stable under pullback in general.

Let $$I := {0to1}$$ be the walking arrow category, then $$1:*to I$$ picks out the terminal object and is thus final. The diagram $$require{AMScd}$$ $$begin{CD} varnothing @>>> *\ @V{F}VV @VV{1}V\ * @>>{0}> I end{CD}$$ is a pullback square, but the functor $$F:varnothingto*$$ is not final. Indeed, let $$G:*tomathbf{Set}$$ pick any nonempty set $$X$$, then $$varinjlim G=X$$, but $$varinjlim Gcirc F=varnothing$$ since the colimit of the empty diagram is just the initial object in $$mathbf{Set}$$. As the unique map $$varnothingto X$$ is not bijective, we can conclude that $$F$$ is not final despite the finality of $$1:*to I$$.

However, final functors form an orthogonal factorisation system with discrete fibrations, and so in particular are closed under pushouts in $$mathbf{Cat}$$ (see e.g., here)

Answered by shibai on November 12, 2021

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