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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJun 1st 2017
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   Author: Mike Shulman
   Format: MarkdownItex[[differential cohesive (infinity,1)-topos]] says > where $i_!$ is a [[full and faithful (∞,1)-functor|full and faithful]] and preserves [[finite products]]. > Equivalently this means that $(i_! \dashv i^\ast) \colon \mathbf{H}_{th} \longrightarrow \mathbf{H}$ is a [[local geometric morphism]] with a further [[right adjoint]] to the right adjoint to the [[direct image]]. But for $(i_! \dashv i^\ast)$ to be a geometric morphism, $i_!$ would need to preserve all finite *limits*?

   differential cohesive (infinity,1)-topos says

   where $i_!$ is a full and faithful and preserves finite products.

   Equivalently this means that $(i_! \dashv i^\ast) \colon \mathbf{H}_{th} \longrightarrow \mathbf{H}$ is a local geometric morphism with a further right adjoint to the right adjoint to the direct image.

   But for $(i_! \dashv i^\ast)$ to be a geometric morphism, $i_!$ would need to preserve all finite limits?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJun 2nd 2017
   • (edited Jun 2nd 2017)
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIf we write $p_! \dashv p^* \dashv p_* \dashv p^!$ for the adjoint string exhibiting cohesion of $\mathbf{H}$, then the composite adjunction $(i_* p^*, p_* i^!) : H_{th} \to \infty Gpd$ is a geometric morphism, hence the unique global sections geometric morphism. If $i_!$ were left exact, then the composite adjunction $(i_! p^*, p_* i^*): H_{th} \to \infty Gpd$ would also be a geometric morphism, hence also the unique global sections geometric morphism, and so we would have $p_* i^! = p_* i^*$ and $i_! p^* = i_* p^*$. The first would mean that the global sections of the infinitesimal shape and flat of something coincide (internally, something like $\sharp\Im X = \sharp 🙰X$), and the second would mean that the reduced and coreduced versions of a discrete object coincide (internally, something like $\Re\flat X = \Im\flat X$). I'm guessing this is not the case in general, but I don't have any intuition for why not. How can discrete cohesion determine nonconstant infinitesimal directions?

   If we write $p_! \dashv p^* \dashv p_* \dashv p^!$ for the adjoint string exhibiting cohesion of $\mathbf{H}$, then the composite adjunction $(i_* p^*, p_* i^!) : H_{th} \to \infty Gpd$ is a geometric morphism, hence the unique global sections geometric morphism. If $i_!$ were left exact, then the composite adjunction $(i_! p^*, p_* i^*): H_{th} \to \infty Gpd$ would also be a geometric morphism, hence also the unique global sections geometric morphism, and so we would have $p_* i^! = p_* i^*$ and $i_! p^* = i_* p^*$. The first would mean that the global sections of the infinitesimal shape and flat of something coincide (internally, something like $\sharp\Im X = \sharp 🙰X$), and the second would mean that the reduced and coreduced versions of a discrete object coincide (internally, something like $\Re\flat X = \Im\flat X$). I’m guessing this is not the case in general, but I don’t have any intuition for why not. How can discrete cohesion determine nonconstant infinitesimal directions?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 2nd 2017
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   Author: Mike Shulman
   Format: MarkdownItexI guess related to this is the question of how the notions of "discrete" and "codiscrete", and their associated modalities, are related in $\mathbf{H}$ and $\mathbf{H}_{th}$. Since $(i_* p^*, p_* i^!)$ is the global sections of $\mathbf{H}_{th}$, it follows that the $\flat$ of $\mathbf{H}_{th}$ is $i_* p^* p_* i^!$, which is the $\flat$ of $\mathbf{H}$ bracketed by infinitesimal flat and coreduction. But the only manifestation I see in $\mathbf{H}_{th}$ of the $\sharp$ of $\mathbf{H}$ is the monad $i_* p^! p_* i^*$, which I think is right adjoint to $\Re \flat\Im$ on $\mathbf{H}_{th}$, but which I don't see any way to express in terms of the six modalities $ʃ,\flat,\sharp,\Re,\Im,\&$ of $\mathbf{H}_{th}$ (since none of them involve $p^!$). And the $\sharp$ of $\mathbf{H}_{th}$ doesn't seem to be related to $\mathbf{H}$ at all. (This is sort of a relaying of a question from Max New, with my own musings around it.)

   I guess related to this is the question of how the notions of “discrete” and “codiscrete”, and their associated modalities, are related in $\mathbf{H}$ and $\mathbf{H}_{th}$. Since $(i_* p^*, p_* i^!)$ is the global sections of $\mathbf{H}_{th}$, it follows that the $\flat$ of $\mathbf{H}_{th}$ is $i_* p^* p_* i^!$, which is the $\flat$ of $\mathbf{H}$ bracketed by infinitesimal flat and coreduction. But the only manifestation I see in $\mathbf{H}_{th}$ of the $\sharp$ of $\mathbf{H}$ is the monad $i_* p^! p_* i^*$, which I think is right adjoint to $\Re \flat\Im$ on $\mathbf{H}_{th}$, but which I don’t see any way to express in terms of the six modalities $ʃ,\flat,\sharp,\Re,\Im,\&$ of $\mathbf{H}_{th}$ (since none of them involve $p^!$). And the $\sharp$ of $\mathbf{H}_{th}$ doesn’t seem to be related to $\mathbf{H}$ at all. (This is sort of a relaying of a question from Max New, with my own musings around it.)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 2nd 2017
   • (edited Jun 2nd 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Mike. re #1: Right, that's an edit mistake. It puzzled me for a moment how I could have written such a blatant nonsense, but looking at the logs, here is what happened: Originally in [rev 1 on Apr 13 2011](https://ncatlab.org/nlab/revision/differential+cohesive+%28infinity%2C1%29-topos/1) I had just required that $i_!$ preserves the terminal object. Then in [rev 49 on Dec 20 2013](https://ncatlab.org/nlab/revision/diff/differential+cohesive+%28infinity%2C1%29-topos/49) something made me go and add to what had used to be "$i_!$ preserves the terminal object" the clause "... and is in fact left exact". And so I also added that comment saying that this means that it is the left adjoint of a local geometric morphism. Then in [rev 73 on Jan 6 2015](https://ncatlab.org/nlab/revision/diff/differential+cohesive+%28infinity%2C1%29-topos/73) I removed that clause again and settled for "$i_!$ preserves finite products" But, as the records show, I forgot to also remove this extra remark, which by that further edit had become obsolete and in fact nonsensical. I have removed it now. Thanks for catching this. Regarding #3: That's right, I have not considered integrating the sharp on $\mathbf{H}$ into the axiomatics in terms of endofunctors on $\mathbf{H}_{th}$. It's great that you are looking into this now. You'll probably discover various improvements. For other readers: we are talking about this diagram: $$ \array{ & && &\overset{i_!}{\hookrightarrow}& \\ & &\overset{p_!}{\longleftarrow}& &\overset{i^\ast}{\longleftarrow}& \\ \Delta \colon & \infty Grpd &\overset{p^\ast}{\hookrightarrow}& \mathbf{H} &\overset{i_\ast}{\hookrightarrow}& \mathbf{H}_{th} \\ & &\overset{p_\ast}{\longleftarrow}& &\overset{i^!}{\longleftarrow}& \\ & &\overset{p^!}{\hookrightarrow}& } $$

   Thanks, Mike.

   re #1: Right, that’s an edit mistake. It puzzled me for a moment how I could have written such a blatant nonsense, but looking at the logs, here is what happened:

   Originally in rev 1 on Apr 13 2011 I had just required that $i_!$ preserves the terminal object.

   Then in rev 49 on Dec 20 2013 something made me go and add to what had used to be "$i_!$ preserves the terminal object" the clause "… and is in fact left exact". And so I also added that comment saying that this means that it is the left adjoint of a local geometric morphism.

   Then in rev 73 on Jan 6 2015 I removed that clause again and settled for "$i_!$ preserves finite products" But, as the records show, I forgot to also remove this extra remark, which by that further edit had become obsolete and in fact nonsensical.

   I have removed it now. Thanks for catching this.

   Regarding #3:

   That’s right, I have not considered integrating the sharp on $\mathbf{H}$ into the axiomatics in terms of endofunctors on $\mathbf{H}_{th}$.

   It’s great that you are looking into this now. You’ll probably discover various improvements.

   For other readers: we are talking about this diagram:

   $$\array{ & && &\overset{i_!}{\hookrightarrow}& \\ & &\overset{p_!}{\longleftarrow}& &\overset{i^\ast}{\longleftarrow}& \\ \Delta \colon & \infty Grpd &\overset{p^\ast}{\hookrightarrow}& \mathbf{H} &\overset{i_\ast}{\hookrightarrow}& \mathbf{H}_{th} \\ & &\overset{p_\ast}{\longleftarrow}& &\overset{i^!}{\longleftarrow}& \\ & &\overset{p^!}{\hookrightarrow}& }$$
5. • CommentRowNumber5.
   • CommentAuthormaxsnew
   • CommentTimeJun 6th 2017
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   Author: maxsnew
   Format: MarkdownItexI added a line to [[Reduction Modality]] to say that $\Re$ preserves finite products.

   I added a line to Reduction Modality to say that $\Re$ preserves finite products.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJun 8th 2017
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   Author: Mike Shulman
   Format: MarkdownItexSo what about the composite $i_! p^*$; is that left exact? Equivalently, does $i_! p^* = i_* p^*$ and $p_*i^* = p_* i^!$?

   So what about the composite $i_! p^*$; is that left exact? Equivalently, does $i_! p^* = i_* p^*$ and $p_*i^* = p_* i^!$?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJun 9th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes. $i_\ast p^\ast = \Delta$ takes an $\infty$-groupoid $X$ to the stack constant on $X$ and since $X \simeq \underset{\longrightarrow}{\lim}_X \ast$ and since $i_!(\ast) \simeq \ast$ the same is true for $i_! p^\ast$.

   Yes.

   $i_\ast p^\ast = \Delta$ takes an $\infty$-groupoid $X$ to the stack constant on $X$ and since $X \simeq \underset{\longrightarrow}{\lim}_X \ast$ and since $i_!(\ast) \simeq \ast$ the same is true for $i_! p^\ast$.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJun 9th 2017
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   Author: Mike Shulman
   Format: MarkdownItexAh, I see. More abstractly, $i_* p^*$ and $i_! p^*$ are both cocontinuous since they are left adjoints, and both preserve the terminal object, but $\infty Gpd$ is the free cocompletion of its terminal object, so $i_* p^* = i_! p^*$. Presumably if $\infty Gpd$ is replaced by some arbitrary base $(\infty,1)$-topos, then an analogous argument would work as long as all the adjunctions are indexed over that base. Thanks!

   Ah, I see. More abstractly, $i_* p^*$ and $i_! p^*$ are both cocontinuous since they are left adjoints, and both preserve the terminal object, but $\infty Gpd$ is the free cocompletion of its terminal object, so $i_* p^* = i_! p^*$. Presumably if $\infty Gpd$ is replaced by some arbitrary base $(\infty,1)$-topos, then an analogous argument would work as long as all the adjunctions are indexed over that base. Thanks!