I have always heard the affine line defined as an R-torsor, and never seen an alternative characterization. I don’t know the alternative axiomatization you are referring to. I would be interested to hear it and see if it does not secretly rely on a very similar and simpler axiomatization of (R,+) itself.
I do know how to characterize the affine line as a topological space without reference to the real numbers.
Torsors seem interesting from the point of view of Occam’s razor because they have less structure but take more words to define.
I do know how to characterize the affine line as a topological space without reference to the real numbers.
This is what I was referring to. The axioms of ordered geometry, especially Dedekind’s axiom, give you the topology of the affine line without a distinguished 0, without distinguishing a direction as “positive”, and without the additive structure.
However, in all the ways I know of to construct a structure satisfying these axioms, you first have to construct the rationals as an ordered field, and the result of course is just the reals, so I don’t know of a constructive way to get at the affine line without constructing the reals with all of their additional field structure.
You might be able to do it with some abstract nonsense. I think general machinery will prove that in categories such as that defined in the top answer of
I have always heard the affine line defined as an R-torsor, and never seen an alternative characterization. I don’t know the alternative axiomatization you are referring to. I would be interested to hear it and see if it does not secretly rely on a very similar and simpler axiomatization of (R,+) itself.
I do know how to characterize the affine line as a topological space without reference to the real numbers.
Torsors seem interesting from the point of view of Occam’s razor because they have less structure but take more words to define.
This is what I was referring to. The axioms of ordered geometry, especially Dedekind’s axiom, give you the topology of the affine line without a distinguished 0, without distinguishing a direction as “positive”, and without the additive structure.
However, in all the ways I know of to construct a structure satisfying these axioms, you first have to construct the rationals as an ordered field, and the result of course is just the reals, so I don’t know of a constructive way to get at the affine line without constructing the reals with all of their additional field structure.
You might be able to do it with some abstract nonsense. I think general machinery will prove that in categories such as that defined in the top answer of
http://mathoverflow.net/questions/92206/what-properties-make-0-1-a-good-candidate-for-defining-fundamental-groups
there are terminal objects. I don’t have time to really think it through though.