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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 $\mathbf{H}$ 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?

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 “$f$ is a tiny object in $\mathcal{H}_{/ Y}$”. 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 $\Im$ 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?

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 $X \to \Im X$ over a point $x \colon \ast \to X \to \Im X$ is the infinitesimal disk around $x$ in $X$.

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 $p$-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 $(W_n)_\ast$.

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 $k$ 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 $Spec(\mathbb{Z}_p)$.

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.