Ну и впрямь бета-версия
Does that mean that a plane from the point of view of Euclidean geometry is not the same as a plane from the point of view of projective or affine geometry? Yes. These are of different types, because they have different notions of identification, and thus they have different properties.
Here we follow Quine’s dictum: No entity without identity! To know a type of objects is to know what it means to identify representatives of the type. The collection of self-identifications (self-transformations) of a given object form a group.
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Groupoids vs groups. The type of all squares in a euclidean plane form a groupoid. It is connected, because between any two there exist identifications between them. But there is no canonical identification. When we say “the symmetry group of the square”, we can mean two things: 1) the symmetry group of a particular square; this is indeed a group, or 2) the connected groupoid of all squares; this is a “group up to conjugation”.
Vector spaces. Constructions and fields. Descartes and cartesian geometry.
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All of mathematics is a tale, not about groups, but about ∞-groupoids. However, a lot of the action happens already with groups.