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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 26th 2015
   • (edited May 26th 2015)
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   Author: Urs
   Format: MarkdownItexthis 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?

   this must be some general abstract fact and I am just being ignorant:

   Consider a topos and a map all whose fibers are tiny objects. Then does the base change (dependent product) preserve colimits?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeMay 26th 2015
   • (edited May 26th 2015)
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   Author: Zhen Lin
   Format: MarkdownItexWhat 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.

   What do you mean by “fibres are tiny objects”? My definition would be “ is a tiny object in ”. After all, a general topos does not have enough points.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 26th 2015
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   Author: Urs
   Format: MarkdownItexRight, 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?

   Right, so I should be more concrete. Take the Cahiers topos, consider the monad which is right adjoint to the comonad which is left Kan extension along the functor that contracts away infinitesimal thickening. Then does dependent product along the units of preserve colimits?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMay 26th 2015
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   Author: Mike Shulman
   Format: MarkdownItexIn what sense are "the fibers" of those maps tiny?

   In what sense are “the fibers” of those maps tiny?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 27th 2015
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   Author: Urs
   Format: MarkdownItexThe fiber of $X \to \Im X$ over a point $x \colon \ast \to X \to \Im X$ is the infinitesimal disk around $x$ in $X$.

   The fiber of over a point is the infinitesimal disk around in .

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 27th 2015
   • (edited May 27th 2015)
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   Author: David_Corfield
   Format: MarkdownItexDon'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](http://www.lorentzcenter.nl/lc/web/2009/342/presentations/lectures%20A.%20Buium.pdf)? Then I should also be thinking about Borger's $(W_n)_\ast$.

   Don’t want to derail this with my current obsession, but in view of #4, where at arithmetic jet space there is

   … the -formal neighbourhood of any arithmetic scheme around a global point is the space of lifts

   this means there should be an such that fiber of is equal to that space of lifts? And that is Buium’s from p.4 here?

   Then I should also be thinking about Borger’s .

7. • CommentRowNumber7.
   • CommentAuthorZhen Lin
   • CommentTimeMay 27th 2015
   • (edited May 27th 2015)
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   Author: Zhen Lin
   Format: MarkdownItex@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.

   @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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 27th 2015
   • (edited May 27th 2015)
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   Author: Urs
   Format: MarkdownItex@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 $Spec(\mathbb{Z}_p)$.

   @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 and consider the unit of .

   @David, yes, this is precisely the incarnation of the formal disk in this arithmetic context, the one around the point in .

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMay 27th 2015
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   Author: Mike Shulman
   Format: MarkdownItexOk, 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.

   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.