axioms for the natural numbers

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

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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