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 definitions 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum 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.
   • CommentAuthorTobyBartels
   • CommentTimeAug 26th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItex[[principle of omniscience]], [[decidable proposition]], [[false]]

   principle of omniscience, decidable proposition, false

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeAug 27th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexMore on [[decidable proposition]]: a terminology change, and categorial interpretation

   More on decidable proposition: a terminology change, and categorial interpretation

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeAug 27th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> In terms of the internal logic of a topos, false is the bottom element in the poset of subobjects of the terminal object. Isn't that more in terms of the *external* logic of the topos? Wouldn't the *internal* meaning of false be the bottom element of the subobject classifier regarded as an internal poset? Maybe this is splitting hairs, but we (or at least I) often seem to get confused on points like that.

   In terms of the internal logic of a topos, false is the bottom element in the poset of subobjects of the terminal object.

   Isn’t that more in terms of the external logic of the topos? Wouldn’t the internal meaning of false be the bottom element of the subobject classifier regarded as an internal poset? Maybe this is splitting hairs, but we (or at least I) often seem to get confused on points like that.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeAug 27th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI basically copied [[false]] from [[true]], mutatis mutandis. So everything (well, almost everything) to be said about one should apply to the other. Anyway, I'll work through this. In the internal logic of a given category, the objects are contexts, and the propositions in a given context $X$ are the subobjects of the object $X$. So the propositions in the global context, which are the propositions with no free variables at all, are the subterminal objects. Then $false$ and $true$ are two of these. However, you can speak of $false$ and $true$ in *any* context, so perhaps we should discuss the bottom/top element in *any* subobject poset. Assuming that we're in a topos (which after all is what was written), subterminal objects correspond to global elements of the subobject classifier $\Omega$, and subobjects of $X$ correspond to morphisms from $X$ to $\Omega$. If we regard $\Omega$ as an internal poset, then the elements of that in the context $X$ are precisely the morphisms from $X$ to the object $\Omega$. Which is what I just said. So you're saying the same as I did in my ‘However, […]’ paragraph. OK, I'll change it.

   I basically copied false from true, mutatis mutandis. So everything (well, almost everything) to be said about one should apply to the other. Anyway, I’ll work through this.

   In the internal logic of a given category, the objects are contexts, and the propositions in a given context $X$ are the subobjects of the object $X$. So the propositions in the global context, which are the propositions with no free variables at all, are the subterminal objects. Then $false$ and $true$ are two of these.

   However, you can speak of $false$ and $true$ in any context, so perhaps we should discuss the bottom/top element in any subobject poset.

   Assuming that we’re in a topos (which after all is what was written), subterminal objects correspond to global elements of the subobject classifier $\Omega$, and subobjects of $X$ correspond to morphisms from $X$ to $\Omega$. If we regard $\Omega$ as an internal poset, then the elements of that in the context $X$ are precisely the morphisms from $X$ to the object $\Omega$. Which is what I just said.

   So you’re saying the same as I did in my ‘However, […]’ paragraph. OK, I’ll change it.

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeAug 27th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexWhile doing this, I had cause to write [[empty category]] and [[poset of subobjects]].

   While doing this, I had cause to write empty category and poset of subobjects.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeAug 27th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexOkay, sure.

   Okay, sure.