Re: axioms for the natural numbers

["George Janelidze" <janelg@telkomsa.net>]

To avoid a possible confusion: I mean 1+1 is the object-of-objects of the
internal category (ordinal) 2 in C (with the rest of structure defined
obviously), not to use 1+1=2 in Cat(C) of course.

--------------------------------------------------
From: "George Janelidze" <janelg@telkomsa.net>
Sent: Monday, August 08, 2011 12:50 AM
To: "Ronnie Brown" <ronnie.profbrown@btinternet.com>; <categories@mta.ca>
Subject: Re: categories: axioms for the natural numbers

> Dear Ronnie,
>
> When the category is, say, lextensive, your way of defining N and Z (and
> thinking of 1+1 as 2) is same as to define them, respectively, as the free
> monoid and the free group on 1. In the case of a topos it well known that
> it is equivalent to Bill's definition (in fact cartesian closedness is
> relevant).
>
> Warm regards
>
> George
>
> --------------------------------------------------
> From: "Ronnie Brown" <ronnie.profbrown@btinternet.com>
> Sent: Friday, August 05, 2011 11:12 PM
> To: <categories@mta.ca>
> Subject: categories: axioms for the natural numbers
>
>> I am aware of the notion of natural number object, based on Bill
>> Lawvere's formulation of induction.
>>
>> But curiously in the category of Sets the natural numbers can be defined
>> as formed from the category 2 (with two objects 0,1 and one arrow from 0
>> to 1) by identifying 0 and 1 in the category of small categories.  This
>> identification can be formulated simply as a pushout in Cat. Using the
>> analogous groupoid I one gets the integers Z - this is one `explanation'
>> of why the fundamental group of the circle is the integers.
>>
>> My question is whether there are any general implications of this kind
>> of `definition' of the natural numbers? Is it, or can it be formulated
>> so as to be,  equivalent to the usual definition, in general situations?
>> Has this been looked at?
>>
>> Ronnie
>>
>>

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