Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2014
   • PermaLink
   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
   • PermaLink
   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
   • PermaLink
   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 $S^1/\mathbb{Z[R$ 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)
   • PermaLink
   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
   • PermaLink
   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
   • PermaLink
   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)
   • PermaLink
   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)
   • PermaLink
   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
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThanks, Urs.

   Thanks, Urs.