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1. • CommentRowNumber1.
   • CommentAuthorMadeleine Birchfield
   • CommentTimeJul 25th 2023
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   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
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   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
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   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.