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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2014
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   Author: Mike Shulman
   Format: MarkdownItexI have added to [[SimpSet]] a list of a few properties of the internal logic of the 1-topos of simplicial sets.

   I have added to SimpSet a list of a few properties of the internal logic of the 1-topos of simplicial sets.

2. • CommentRowNumber2.
   • CommentAuthorspitters
   • CommentTimeMar 12th 2014
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   Author: spitters
   Format: MarkdownItexNice! Dependent choice holds in any presheaf topos (I added a reference). It would be nice to have a parallel page for cubical sets.

   Nice! Dependent choice holds in any presheaf topos (I added a reference).

   It would be nice to have a parallel page for cubical sets.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 15th 2014
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   Author: Todd_Trimble
   Format: MarkdownItexI can't see the paper by Fourman and Scedrov, but the statement that dependent choice holds in any presheaf topos needs to be qualified. There is relevant material on this at [[presentation axiom]]. The external statement, that for every entire relation $R$ on an inhabited set $X$ there exists a sequence $x_{-}: N \to X$ such that $x_n R x_{n+1}$, clearly fails if there are inhabited sets $X$ which have no global elements $x_0: 1 \to X$ (i.e., if $1$ is not [externally] projective). I'm not sure about internal analogues, but insofar as the internal form of the presentation axiom may fail for some presheaf toposes, I'm somewhat skeptical. In the presence of the external presentation axiom (which holds for all presheaf toposes $[C^{op}, Set]$), projectivity of $1$ is equivalent to DC. If $C$ has a terminal object $t$, then $1 = C(-, t)$ is projective (as any representable must be), and this is certainly the case for $C = \Delta$.

   I can’t see the paper by Fourman and Scedrov, but the statement that dependent choice holds in any presheaf topos needs to be qualified. There is relevant material on this at presentation axiom. The external statement, that for every entire relation $R$ on an inhabited set $X$ there exists a sequence $x_{-}: N \to X$ such that $x_n R x_{n+1}$, clearly fails if there are inhabited sets $X$ which have no global elements $x_0: 1 \to X$ (i.e., if $1$ is not [externally] projective). I’m not sure about internal analogues, but insofar as the internal form of the presentation axiom may fail for some presheaf toposes, I’m somewhat skeptical.

   In the presence of the external presentation axiom (which holds for all presheaf toposes $[C^{op}, Set]$), projectivity of $1$ is equivalent to DC. If $C$ has a terminal object $t$, then $1 = C(-, t)$ is projective (as any representable must be), and this is certainly the case for $C = \Delta$.

4. • CommentRowNumber4.
   • CommentAuthorspitters
   • CommentTimeMar 15th 2014
   • (edited Mar 15th 2014)
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   Author: spitters
   Format: MarkdownItexHere is the [PDF](http://link.springer.com/content/pdf/10.1007%2FBF01170929.pdf). In fact, IIRC it should be go both ways. DC holds is $Set$ iff it holds in one, and hence all, presheaf categories.

   Here is the PDF. In fact, IIRC it should be go both ways. DC holds is $Set$ iff it holds in one, and hence all, presheaf categories.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 15th 2014
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   Author: Todd_Trimble
   Format: MarkdownItexThe link doesn't send me to the article. But do you follow my objection? I thought I expressed it pretty clearly.

   The link doesn’t send me to the article. But do you follow my objection? I thought I expressed it pretty clearly.

6. • CommentRowNumber6.
   • CommentAuthorspitters
   • CommentTimeMar 15th 2014
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   Author: spitters
   Format: MarkdownItexSorry, there was a / missing, I editted the link above. The result in the paper is about the internal version of DC (Kripke-Joyal semantics). I am probably slow, why should projectivity of 1 be equivalent to DC? I can imagine how it implies it.

   Sorry, there was a / missing, I editted the link above.

   The result in the paper is about the internal version of DC (Kripke-Joyal semantics).

   I am probably slow, why should projectivity of 1 be equivalent to DC? I can imagine how it implies it.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 15th 2014
   • (edited Mar 15th 2014)
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   Author: Todd_Trimble
   Format: MarkdownItexThe necessity of projectivity of $1$ is just as I explained in #3, although I was referring to the external version of DC, which speaks of an actual morphism $\mathbb{N} \to X$. But let me have a look at the paper (and thanks). Probably the only qualification needed is a matter of mumbling something about the _internal_ version of DC, but let me have a look. Edit: Argh, problem is that the article is behind a pay wall, and I don't have institutional access to it.

   The necessity of projectivity of $1$ is just as I explained in #3, although I was referring to the external version of DC, which speaks of an actual morphism $\mathbb{N} \to X$. But let me have a look at the paper (and thanks). Probably the only qualification needed is a matter of mumbling something about the internal version of DC, but let me have a look.

   Edit: Argh, problem is that the article is behind a pay wall, and I don’t have institutional access to it.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMar 15th 2014
   • (edited Mar 15th 2014)
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   Author: Urs
   Format: MarkdownItexTodd, if you google for the title of the article, the fifth hit or so (EUDML) takes you to an online readable version. (Would post the link, but cannot from my phone here.)

   Todd, if you google for the title of the article, the fifth hit or so (EUDML) takes you to an online readable version. (Would post the link, but cannot from my phone here.)

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 16th 2014
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   Author: Todd_Trimble
   Format: MarkdownItexThanks, Urs.

   Thanks, Urs.