axioms for the natural numbers

axioms for the natural numbers

[Ronnie Brown]

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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Re: axioms for the natural numbers

[George Janelidze]

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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