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this must be some general abstract fact and I am just being ignorant:
Consider a topos $\mathbf{H}$ and a map $f \colon X \longrightarrow Y$ all whose fibers are tiny objects. Then does the base change $f_\ast = \prod_f \colon \mathbf{H}_{/X} \longrightarrow \mathbf{H}_{/Y}$ (dependent product) preserve colimits?
What do you mean by “fibres are tiny objects”? My definition would be “$f$ is a tiny object in $\mathcal{H}_{/ Y}$”. After all, a general topos does not have enough points.
Right, so I should be more concrete. Take the Cahiers topos, consider the monad $\Im$ which is right adjoint to the comonad $\Re$ which is left Kan extension along the functor that contracts away infinitesimal thickening. Then does dependent product along the units $X \to \Im X$ of $\Im$ preserve colimits?
In what sense are “the fibers” of those maps tiny?
The fiber of $X \to \Im X$ over a point $x \colon \ast \to X \to \Im X$ is the infinitesimal disk around $x$ in $X$.
Don’t want to derail this with my current obsession, but in view of #4, where at arithmetic jet space there is
… the $p$-formal neighbourhood of any arithmetic scheme $X$ around a global point $x \colon Spec(\mathbb{Z}) \to X$ is the space of lifts
$\array{ Spf(\mathbb{Z}_p) && \stackrel{\hat x}{\longrightarrow}&& X \\ & \searrow && \swarrow \\ && Spec(\mathbb{Z}) } \,,$
this means there should be an $\Im$ such that fiber of $X \to \Im(X)$ is equal to that space of lifts? And that $\Im(X)$ is Buium’s $J_\infty$ from p.4 here?
Then I should also be thinking about Borger’s $(W_n)_\ast$.
@Urs
I’m not really familiar with the Cahiers topos, but are those infinitesimal discs really tiny objects? My impression is that they are not representable sheaves (but perhaps rather ind-representable), which seems like an obstacle to being tiny.
@Zhen Lin: true, if we allow infinitesimals of arbitrary order, then those infinitesimal disks are genuine formal disks. I am happy to restrict attention to the case where we use only infinitesimals up to any fixed finite order $k$ and consider the unit of $X \longrightarrow \Im_{(k)}X$.
@David, yes, this $Spf(\mathbb{Z}_p)$ is precisely the incarnation of the formal disk $\mathbb{D}$ in this arithmetic context, the one around the point $(p)$ in $Spec(\mathbb{Z}_p)$.
Ok, well, as Zhen suggested in #2, we should probably expect to need a more internal notion of “tiny fibers” rather than just looking at the global points.
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