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1. • CommentRowNumber1.
   • CommentAuthorThomas Holder
   • CommentTimeJul 18th 2015
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   Author: Thomas Holder
   Format: MarkdownItexThis is just a pointer to the recent tac paper from Lawvere and Menni on "[Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness](http://www.tac.mta.ca/tac/volumes/30/26/30-26abs.html)".

   This is just a pointer to the recent tac paper from Lawvere and Menni on “Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness”.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 18th 2015
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   Author: Urs
   Format: MarkdownItexthanks. I have added the pointer * [[William Lawvere]], [[Matías Menni]], _Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness_, Theory and Applications of Categories, Vol. 30, 2015, No. 26, pp 909-932. ([TAC](http://www.tac.mta.ca/tac/volumes/30/26/30-26abs.html)) to _[[double negation topology]]_, to _[[cohesive topos]]_ and to _[[Aufhebung]]_

   thanks. I have added the pointer

   • William Lawvere, Matías Menni, Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness, Theory and Applications of Categories, Vol. 30, 2015, No. 26, pp 909-932. (TAC)

   to double negation topology, to cohesive topos and to Aufhebung

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJul 19th 2015
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   Author: Mike Shulman
   Format: MarkdownItexThanks. I would like to understand this paper, but right now I can't make much of it: it reads like a long string of unmotivated technical definitions. If anyone has the time to make their way through it and write a gloss that includes motivation and examples, I would enjoy reading it.

   Thanks. I would like to understand this paper, but right now I can’t make much of it: it reads like a long string of unmotivated technical definitions. If anyone has the time to make their way through it and write a gloss that includes motivation and examples, I would enjoy reading it.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 19th 2015
   • (edited Jul 19th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexNot sure if you saw the nLab paragraph I added before the Lab went down, this is the point that concerns what we had discussed here before re double negation and it culminates in 4.1-4.4: 1) pieces-have-points implies initial Aufhebung $\sharp \emptyset \simeq \emptyset$, 2) the converse follows if the $\sharp$-base is Boolean and hence in summary 3) for cohesion over a Boolean base, pieces-have-points is equivalent to $\sharp$ being double negation localization.

   Not sure if you saw the nLab paragraph I added before the Lab went down, this is the point that concerns what we had discussed here before re double negation and it culminates in 4.1-4.4:

   1) pieces-have-points implies initial Aufhebung $\sharp \emptyset \simeq \emptyset$,

   2) the converse follows if the $\sharp$-base is Boolean

   and hence in summary

   3) for cohesion over a Boolean base, pieces-have-points is equivalent to $\sharp$ being double negation localization.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJul 20th 2015
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   Author: Mike Shulman
   Format: MarkdownItexParagraph on what page?

   Paragraph on what page?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 20th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexSorry for being unclear, I had announced that in a [separate thread](http://nforum.ncatlab.org/discussion/6681/pieceshavepoints-is-initial-resolution/?Focus=53937#Comment_53937). But it's just a tiny remark that I had added, now we have already spent more keystrokes on it than it may be worth it: I have added the statement of lemmas 4.1, 4.2 of Menni-Lawvere to _[[cohesive topos]]_ [here](http://ncatlab.org/nlab/show/cohesive+topos#RelationOfPiecesToPointsToInitialAufhebung) and to _[[points-to-pieces transform]]_ [here](http://ncatlab.org/nlab/show/points-to-pieces%20transform#RelationToAufhebung).

   Sorry for being unclear, I had announced that in a separate thread. But it’s just a tiny remark that I had added, now we have already spent more keystrokes on it than it may be worth it:

   I have added the statement of lemmas 4.1, 4.2 of Menni-Lawvere to cohesive topos here and to points-to-pieces transform here.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJul 21st 2015
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   Author: Mike Shulman
   Format: MarkdownItexAh, thanks. That helps me know what to look for, at least.

   Ah, thanks. That helps me know what to look for, at least.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJul 21st 2015
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIs this true in the ∞-case too?

   Is this true in the ∞-case too?

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJul 21st 2015
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think it is. This is very nice! It means that for classical real-cohesion I don't need to assume $\sharp\emptyset=\emptyset$ as an additional axiom. And it's intriguing how we can use ʃ to give us information about $\sharp$.

   I think it is. This is very nice! It means that for classical real-cohesion I don’t need to assume $\sharp\emptyset=\emptyset$ as an additional axiom. And it’s intriguing how we can use ʃ to give us information about $\sharp$.

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJul 21st 2015
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexHere's another question: what can we say about the composite $ʃ\circ \sharp$?

    Here’s another question: what can we say about the composite $ʃ\circ \sharp$?

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2015
    • (edited Jul 21st 2015)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> I think it is. That's neat. Unfortunately I have no leisure for this right now. So what exactly do you think goes through? The full higher analog of both lemmas? > what can we say about the composite ʃ∘♯? While I don't know, intuitively I'd expect this to be contractible. In smooth cohesion it is known now that ʃX is given by $\underset{\rightarrow}{\lim}[\Delta_{smth}^\bullet, X]$. And I'd expect that $\underset{\rightarrow}{\lim}[\Delta_{smth}^\bullet, \sharp X] \simeq \ast$. But I haven't really thought about it any further.

    I think it is.

    That’s neat. Unfortunately I have no leisure for this right now. So what exactly do you think goes through? The full higher analog of both lemmas?

    what can we say about the composite ʃ∘♯?

    While I don’t know, intuitively I’d expect this to be contractible. In smooth cohesion it is known now that ʃX is given by $\underset{\rightarrow}{\lim}[\Delta_{smth}^\bullet, X]$. And I’d expect that $\underset{\rightarrow}{\lim}[\Delta_{smth}^\bullet, \sharp X] \simeq \ast$. But I haven’t really thought about it any further.

12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJul 21st 2015
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    Author: Mike Shulman
    Format: MarkdownItexLemma 4.1 makes perfect sense higher categorically if by "Nullstellensatz" one means pieces have points. I think Lemma 4.2 works fine too, but what it proves is that discrete objects are closed under subobjects. The equivalence of those is trickier, if nothing else trickier to track down (their Lemma 3.1 combined with Lemma 2.3 from Johnstone's paper and A4.6.6 from the Elephant), but right now I think it should still be true. I think one can only obtain that $\flat A \to A$ is mono when $A$ is 0-truncated (and one shouldn't expect more), but I think that should be enough since surjectivity is detected by the 0-truncation.

    Lemma 4.1 makes perfect sense higher categorically if by “Nullstellensatz” one means pieces have points. I think Lemma 4.2 works fine too, but what it proves is that discrete objects are closed under subobjects. The equivalence of those is trickier, if nothing else trickier to track down (their Lemma 3.1 combined with Lemma 2.3 from Johnstone’s paper and A4.6.6 from the Elephant), but right now I think it should still be true. I think one can only obtain that $\flat A \to A$ is mono when $A$ is 0-truncated (and one shouldn’t expect more), but I think that should be enough since surjectivity is detected by the 0-truncation.

13. • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJul 21st 2015
    • (edited Jul 22nd 2015)
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI agree that intuitively $ʃ \sharp A$ should be contractible as long as $A$ is inhabited. In general I'd expect $ʃ \sharp A$ to be $\Vert A \Vert$, the propositional truncation. So far, I can prove in real-cohesion that $ʃ \sharp A$ is "conditionally connected", i.e. $\prod_{x,y:ʃ \sharp A} \Vert x=y\Vert$. I think the proof should work in smooth cohesion too; all it needs is that there is an object $R$ with two distinct points such that $ʃR$ is contractible.

    I agree that intuitively $ʃ \sharp A$ should be contractible as long as $A$ is inhabited. In general I’d expect $ʃ \sharp A$ to be $\Vert A \Vert$, the propositional truncation. So far, I can prove in real-cohesion that $ʃ \sharp A$ is “conditionally connected”, i.e. $\prod_{x,y:ʃ \sharp A} \Vert x=y\Vert$. I think the proof should work in smooth cohesion too; all it needs is that there is an object $R$ with two distinct points such that $ʃR$ is contractible.

14. • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJul 22nd 2015
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    Author: Mike Shulman
    Format: MarkdownItexHaha, now I see that this is exactly the other conditions that Lawvere and Menni are studying. Their "connected codiscreteness" (CC) says that $ʃ\sharp A$ is always subterminal, which in the presence of pieces-have-points is equivalent to it being $\Vert A\Vert$. They also prove (due mainly to Johnstone) that it's equivalent to $ʃ\sharp 2 = 1$, but I haven't inspected that proof for ∞-ization.

    Haha, now I see that this is exactly the other conditions that Lawvere and Menni are studying. Their “connected codiscreteness” (CC) says that $ʃ\sharp A$ is always subterminal, which in the presence of pieces-have-points is equivalent to it being $\Vert A\Vert$. They also prove (due mainly to Johnstone) that it’s equivalent to $ʃ\sharp 2 = 1$, but I haven’t inspected that proof for ∞-ization.