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 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 sheaves 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.
   • CommentAuthorMadeleine Birchfield
   • CommentTimeJul 25th 2023
   • PermaLink
   Author: Madeleine Birchfield
   Format: MarkdownItexstarting an article on extended natural numbers objects <a href="https://ncatlab.org/nlab/revision/extended+natural+numbers+object/1">v1</a>, <a href="https://ncatlab.org/nlab/show/extended+natural+numbers+object">current</a>

   starting an article on extended natural numbers objects

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 25th 2023
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexDoes this general notion of extended natural numbers object appear somewhere in the literature?

   Does this general notion of extended natural numbers object appear somewhere in the literature?

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 28th 2023
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI'm reminded of the object Martin Escardo likes using, the type of non-increasing boolean sequences. Classically this is the naturals plus a point at infinity. In topological models it's the one-point compactification of N. But in computational settings it's richer. In particular, the complement of N that sits inside is not the infinite point, but you cannot construct a term that's not either a finite number of infinity. This sees to me to be interesting in other toposes, for instance: in the category of sheaves of a space, is the corresponding object the sheaf of continuous functions to the one-point compactification of N? In the way the Dedekind reals are the real-valued functions, and the Cauchy reals are the locally constant real functions.

   I’m reminded of the object Martin Escardo likes using, the type of non-increasing boolean sequences. Classically this is the naturals plus a point at infinity. In topological models it’s the one-point compactification of N. But in computational settings it’s richer. In particular, the complement of N that sits inside is not the infinite point, but you cannot construct a term that’s not either a finite number of infinity. This sees to me to be interesting in other toposes, for instance: in the category of sheaves of a space, is the corresponding object the sheaf of continuous functions to the one-point compactification of N? In the way the Dedekind reals are the real-valued functions, and the Cauchy reals are the locally constant real functions.