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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeAug 15th 2011

    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 nn-concept from higher category theory makes sense “for any extended natural number nn”.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 16th 2011

    So we could say decategorification and looping (as decribed in Categorification)) send nn-categories to pred(n)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)(-2)-categories).

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeAug 17th 2011

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

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 17th 2011

    I 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.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 17th 2011

    @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 nn-categories are conveniently defined inductively for finite nn and coinductively for infinite nn.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeAug 17th 2011
    • (edited Aug 17th 2011)

    @DR

    Yes, there was supposed to be x ix_i in there; I’ve fixed it. (I also divided by 22 twice for some reason.)

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

    • CommentRowNumber7.
    • CommentAuthorColin Tan
    • CommentTimeNov 26th 2018
    • (edited Nov 26th 2018)

    Should the extended natural numbers start with 11 instead? It is written in the section on universal property, “You can think of corec Spcorec_S p as mapping an element aa of SS to the number of times that pp must be applied in succession, starting from aa, before being taken out of SS.” In particular, if p(a)=*p(a) = * is undefined, so pp must be applied 11 time before being taken out of SS, so corec Sp(a)\corec_S p(a) should be equal to 11, right?

    So it seems that corec S\corec_S takes values 1,2,1, 2, \dots or \infty.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 27th 2018

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

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeNov 27th 2018

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

    diff, v10, current

    • CommentRowNumber10.
    • CommentAuthorRichard Williamson
    • CommentTimeNov 28th 2018
    • (edited Nov 28th 2018)

    [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. ]

    • CommentRowNumber11.
    • CommentAuthorColin Tan
    • CommentTimeNov 28th 2018

    In that sense, the set of extended natural numbers is a decategorification of the category of countable sets. Identify 00 with the empty set, 11 with a singleton, …, \infty with a countable set. And pp identifies with the operation of taking the complement of a singleton in a set. Eg p()p(\emptyset) is undefined, p({0})={0}{0}=p(\{0\}) = \{0\} \setminus \{0\} = \emptyset is empty, p({0,1})={0,1}{1}={1}p(\{0, 1\}) = \{0, 1\} \setminus \{1\} = \{1\} is a singleton, … and p({0,1,2,...})={0,1,2,...}{0}={1,2,}p(\{0, 1, 2, ... \}) = \{0, 1, 2, ...\} \setminus \{0\} = \{1, 2, \dots\} is countable.

    So corec Sp(a)\corec_S p(a) is the extended natural number identified with the subset of {p,pp,ppp,}\{p, p\circ p, p\circ p \circ p, \cdots \} containing those ppp \circ \cdots \circ p that satisfy (pp)(a)S(p \circ \cdots \circ p)(a) \in S. Thanks for suggesting the change, David!

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