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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeAug 15th 2011
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   Author: TobyBartels
   Format: MarkdownItexI moved [[extended natural number system]] to [[extended natural number]] and expanded it. In particular, it's now a good place to link to if you want to say that an $n$-concept from higher category theory makes sense "for any [[extended natural number]] $n$".

   I moved extended natural number system to extended natural number and expanded it. In particular, it’s now a good place to link to if you want to say that an -concept from higher category theory makes sense “for any extended natural number ”.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 16th 2011
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   Author: David_Corfield
   Format: MarkdownItexSo we could say decategorification and looping (as decribed in [Categorification](http://arxiv.org/abs/math/9802029))) send $n$-categories to $pred(n)$-categories for all extended natural numbers (having tidied up numbering at the lower end and doing what is necessary to allow $\ast$-categories to be images of $(-2)$-categories).

   So we could say decategorification and looping (as decribed in Categorification)) send -categories to -categories for all extended natural numbers (having tidied up numbering at the lower end and doing what is necessary to allow -categories to be images of -categories).

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeAug 17th 2011
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   Author: TobyBartels
   Format: MarkdownItexYes! (And looping makes a monoidal $pred(n)$-category.)

   Yes! (And looping makes a monoidal -category.)

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 17th 2011
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   Author: DavidRoberts
   Format: MarkdownItexI have a query on the page about the embedding in $\mathbb{R}$, because even though I made a small fix, I still don't think I got it right.

   I have a query on the page about the embedding in , because even though I made a small fix, I still don’t think I got it right.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeAug 17th 2011
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   Author: Mike Shulman
   Format: MarkdownItex@David C. 2: That's a nice way to put it! I don't know if I put together before the facts that the ordinary natural numbers are the well-founded objects in the extended natural numbers (the inductive type sitting inside the coinductive type), whereas $n$-categories are conveniently defined inductively for finite $n$ and coinductively for infinite $n$.

   @David C. 2: That’s a nice way to put it! I don’t know if I put together before the facts that the ordinary natural numbers are the well-founded objects in the extended natural numbers (the inductive type sitting inside the coinductive type), whereas -categories are conveniently defined inductively for finite and coinductively for infinite .

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeAug 17th 2011
   • (edited Aug 17th 2011)
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   Author: TobyBartels
   Format: MarkdownItex@DR Yes, there was supposed to be $x_i$ in there; I've fixed it. (I also divided by $2$ twice for some reason.) The crazy way that it looked before is another itex "feature".

   @DR

   Yes, there was supposed to be in there; I’ve fixed it. (I also divided by twice for some reason.)

   The crazy way that it looked before is another itex “feature”.

7. • CommentRowNumber7.
   • CommentAuthorColin Tan
   • CommentTimeNov 27th 2018
   • (edited Nov 27th 2018)
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   Author: Colin Tan
   Format: MarkdownItexShould the extended natural numbers start with $1$ instead? It is written in the section on universal property, "You can think of $corec_S p$ as mapping an element $a$ of $S$ to the number of times that $p$ must be applied in succession, starting from $a$, before being taken out of $S$." In particular, if $p(a) = *$ is undefined, so $p$ must be applied $1$ time before being taken out of $S$, so $\corec_S p(a)$ should be equal to $1$, right? So it seems that $\corec_S$ takes values $1, 2, \dots$ or $\infty$.

   Should the extended natural numbers start with instead? It is written in the section on universal property, “You can think of as mapping an element of to the number of times that must be applied in succession, starting from , before being taken out of .” In particular, if is undefined, so must be applied time before being taken out of , so should be equal to , right?

   So it seems that takes values or .

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 27th 2018
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   Author: David_Corfield
   Format: MarkdownItexOr redefine so that the number counts the times that $p$ can be applied while staying within $S$.

   Or redefine so that the number counts the times that can be applied while staying within .

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeNov 27th 2018
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   Author: Mike Shulman
   Format: MarkdownItexI changed the wording to "the maximum number of times that $p$ can be applied in succession, starting from $a$, before being taken out of $S$" to keep 0. <a href="https://ncatlab.org/nlab/revision/diff/extended+natural+number/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/extended+natural+number/10">v10</a>, <a href="https://ncatlab.org/nlab/show/extended+natural+number">current</a>

   I changed the wording to “the maximum number of times that can be applied in succession, starting from , before being taken out of ” to keep 0.

   diff, v10, current

10. • CommentRowNumber10.
    • CommentAuthorRichard Williamson
    • CommentTimeNov 28th 2018
    • (edited Nov 28th 2018)
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    Author: Richard Williamson
    Format: MarkdownItex[Administrative note: comments #1 - #8 originally belonged to a Latest Changes thread entitled 'Extended natural numbers'. The edit announcer does not know about page name changes that were made before it came into existence, so I have manually merged that thread with this one. I also deleted a one sentence comment of Mike's in the old thread which basically just alerted to the fact that the edit announcer had used this thread. ]

    [Administrative note: comments #1 - #8 originally belonged to a Latest Changes thread entitled ’Extended natural numbers’. The edit announcer does not know about page name changes that were made before it came into existence, so I have manually merged that thread with this one. I also deleted a one sentence comment of Mike’s in the old thread which basically just alerted to the fact that the edit announcer had used this thread. ]

11. • CommentRowNumber11.
    • CommentAuthorColin Tan
    • CommentTimeNov 28th 2018
    • PermaLink
    Author: Colin Tan
    Format: MarkdownItexIn that sense, the set of extended natural numbers is a decategorification of the category of countable sets. Identify $0$ with the empty set, $1$ with a singleton, ..., $\infty$ with a countable set. And $p$ identifies with the operation of taking the complement of a singleton in a set. Eg $p(\emptyset)$ is undefined, $p(\{0\}) = \{0\} \setminus \{0\} = \emptyset$ is empty, $p(\{0, 1\}) = \{0, 1\} \setminus \{1\} = \{1\}$ is a singleton, ... and $p(\{0, 1, 2, ... \}) = \{0, 1, 2, ...\} \setminus \{0\} = \{1, 2, \dots\}$ is countable. So $\corec_S p(a)$ is the extended natural number identified with the subset of $\{p, p\circ p, p\circ p \circ p, \cdots \}$ containing those $p \circ \cdots \circ p$ that satisfy $(p \circ \cdots \circ p)(a) \in S$. Thanks for suggesting the change, David!

    In that sense, the set of extended natural numbers is a decategorification of the category of countable sets. Identify with the empty set, with a singleton, …, with a countable set. And identifies with the operation of taking the complement of a singleton in a set. Eg is undefined, is empty, is a singleton, … and is countable.

    So is the extended natural number identified with the subset of containing those that satisfy . Thanks for suggesting the change, David!

12. • CommentRowNumber12.
    • CommentAuthorVictor Sannier
    • CommentTimeDec 16th 2023
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    Author: Victor Sannier
    Format: MarkdownItexAdded redirections of conatural numbers, conat, conumber, etc. and wrote a section on formalisation in proof assistants <a href="https://ncatlab.org/nlab/revision/diff/extended+natural+number/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/extended+natural+number/14">v14</a>, <a href="https://ncatlab.org/nlab/show/extended+natural+number">current</a>

    Added redirections of conatural numbers, conat, conumber, etc. and wrote a section on formalisation in proof assistants

    diff, v14, current