This is primarily motivated from one of the discussions that happened in class.
One can think of the difference between affine space and vector space in another way. Consider a vector space V, this comes with an additive group of the vector space (which I also denote by V). So one defines an affine space A associated to this vector space as the set of "points" on which the group V acts by translation (this is free and transitive), and given two points in the affine space, say p,q , there is a unique vector w of V such that p - q = w
So in a certain sense, this construction of the affine space shows how an affine space forgets the origin and only remembers the difference between two distinct points. A naive motivation would be to think of the affine space as physical points and the vector space associated as direction vectors(maybe velocities?) between points. A concrete concept here is that of G-torsor, where G is a group. An affine space is then a V-torsor
I was wondering if there could be such a similar interpretation of the projective space as some kind of a torsor?