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Stream: learning: questions

Topic: Is evaluation-at-1 a fibration?

Jules Hedges (Feb 21 2026 at 19:10):

Write $\mathbf{End} (\mathbf{Set})$ for the category of $\mathbf{Set}$-endofunctors and $\mathbf{Poly}$ for the full subcategory of polynomials, and let $P : \mathbf{End} (\mathbf{Set}) \to \mathbf{Set}$ be the evaluation-at-1 functor $P (F) = F (1)$. $P$ restricted to $\mathbf{Poly}$ is a fibration, whose fibres are $P^{-1} (X) = (\mathbf{Set} / X)^\mathrm{op}$. I'm trying to figure out whether $P$ on arbitrary endofunctors is also a fibration, and if so, what its fibres are

I have a candidate for cartesian lifting, which has to take a functor $F : \mathbf{Set} \to \mathbf{Set}$ and a function $f : X \to F(1)$ to a functor $f^* (F) : \mathbf{Set} \to \mathbf{Set}$ with the property that $f^* (F) (1) = X$. But there's a lot of details I need to check, so I'm going to ask here in case anybody already knows the answer

For polynomials (ie. coproducts of representables) $F (Y) = \sum_{a : A} Y^{A' (a)}$, cartesian liftings are given by $f^* (F) (Y) = \sum_{x : X} Y^{A' (f(x))}$. So the obvious thing to do is to try to replicate that but for general colimits of representables. As far as I can tell, the way to do this is to take the category of elements $\int F$ and its projection $\pi : \int F \to \mathbf{Set}$, and then $F (Y) \cong \mathrm{colim}_{A : (\int F)^\mathrm{op}} Y^{\pi (A)}$ (where the colimit is over $(\int F)^\mathrm{op}$ because we are talking about copresheaves as colimits of covariant representables instead of the more common version... but I'm not very confident at this bit)

Then colimits of presheaves are computed pointwise, so $F (1) \cong \mathrm{colim}_{A : (\int F)^\mathrm{op}} 1 \cong \pi_0 ((\int F)^\mathrm{op}) \cong \pi_0 (\int F)$, the set of connected components of $\int F$

Next, I convinced myself that the connected components functor $\pi_0 : \mathbf{Cat} \to \mathbf{Set}$ is probably a fibration (but I haven't been able to find any reference for that, which makes me very suspcious). Namely, every category can be canonically decomposed as a coproduct of categories $\mathcal C = \sum_{x : \pi_0 (\mathcal C)} \mathcal C_x$ where each $\mathcal C_x$ is connected, and then for a function $f : X \to \pi_0 (\mathcal C)$ there is an obvious definition of $f^* (\mathcal C) = \sum_{x : X} \mathcal C_{f (x)}$

So, if I have a function $f : X \to F(1) \cong \pi_0 (\int F)$, I can form the category $\mathcal C = f^* ((\int F)^\mathrm{op})$, whose set of connected components is $X$ and which is equipped with a universal functor $H : \mathcal C \to (\int F)^\mathrm{op}$. And with all that I can finally write my candidate definition for cartesian lifting: $f^* (F) (Y) = \mathrm{colim}_{X : \mathcal C} Y^{\pi (H (X))}$

There's potentially many mistakes in all that, and even if everything works I also have no idea where I'd start with calculating the fibres, so I'm dumping it all here just in case somebody already knows the answer

Jules Hedges (Feb 21 2026 at 19:21):

One thing that's easy to notice is that if $P$ is a fibration then its fibres must be locally large, since $\mathbf{End} (\mathbf{Set})$ is. In fact the only way I know how to prove the magical fact that $\mathbf{Poly}$ is locally small is to equip it with the fibration $P$ and noticing that the fibres $(\mathbf{Set} / X)^\mathrm{op}$ are all locally small

Matteo Capucci (he/him) (Feb 23 2026 at 08:25):

I know the answer is yes but I don't have a full write up of that (I have a partial write up in §6 here)

Matteo Capucci (he/him) (Feb 23 2026 at 08:28):

You might want to check Mordeijk and Joyal's paper(s) on open/étale maps. That fibration is related to those, when you write $End(Set) = Set^{Set}$ you see why this is a topos-theoretic concept.

Matteo Capucci (he/him) (Feb 23 2026 at 08:28):

Specifically the functor $Set^C \to Set$ given by evaluation at the terminal object of $C$ has a special property I forgot the name of. Alas...

Jules Hedges (Feb 24 2026 at 12:38):

One thing that makes me slightly suspicious (I'm trying to poke holes before I commit to doing a whole load of calculation) is that $\mathbf{Poly}$ is not a topos (and neither is the smaller full subcategory of functors of the form $A \times Y^B$, for which evaluation-at-1 is also a fibration), so this makes it sound like there's a strange coincidence where evaluation-at-1 can be a fibration for 2 different reasons

Matteo Capucci (he/him) (Feb 24 2026 at 12:50):

uhm I think the reason is the same, the fact it is not a topos is not clearly relevant. when I said topos-theoretic concept I didn't mean that it's exclusive to topoi, but that it is relevant for topoi.

Martti Karvonen (Feb 24 2026 at 17:01):

For polynomials (ie. coproducts of representables) $F (Y) = \sum_{a : A} Y^{A' (a)}$, cartesian liftings are given by $f^* (F) (Y) = \sum_{x : X} Y^{A' (f(x))}$. So the obvious thing to do is to try to replicate that but for general colimits of representables. As far as I can tell, the way to do this is to take the category of elements $\int F$ and its projection $\pi : \int F \to \mathbf{Set}$, and then $F (Y) \cong \mathrm{colim}_{A : (\int F)^\mathrm{op}} Y^{\pi (A)}$ (where the colimit is over $(\int F)^\mathrm{op}$ because we are talking about copresheaves as colimits of covariant representables instead of the more common version... but I'm not very confident at this bit)

As an aside, I'm a bit uneasy about the large limit taken here, but maybe one can make things nicer by restricting to small copreseheaves.