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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