From: Steve Vickers
That seems to require a new notion of torsor, and I can't see how it would work technically.
I don't know if it's a new notion, but the following construction of what I'll call Aff(C), the affinitization (affination?) of C, should work for any category C concrete over Ab via a faithful functor U: C --> Ab. (Ab(Z,-): Ab --> Set automatically makes Ab concrete over Set.) The subcategories of Vct_k we've been talking about, in particular Euc with origin but liberal morphisms, are all instances of such a C. If this is different from torsors I'd appreciate some insight into the difference. Definition: Take the objects of Aff(C) to be those of C. A morphism in Aff(C) from c to d is a pair (h, m) where m: c --> d is a morphism of C and h: U(c) --> U(d) is any group homomorphism such that h(x) - h(y) = U(m)(x - y) for all x, y in U(c). End definition. The effect of this construction is to make the objects of Aff(c) generalized metric spaces. Reading between the lines of the above, an implicit metric D(x, y) is given by the group structure in the form x + D(x, y) = y, or equivalently D(x, y) = y - x; there is no need to make D explicit with its own symbol. All four Frechet axioms follow after modifying them to make the triangle inequality an equality and D(x, y) = -D(y, x) (symmetry becomes antisymmetry). The morphism (h, m) maps the points of the metric space via h subject to coherently (over the whole space) maintaining the distance with m. That should do it for Euc. France Dacar did things in the other order: torsor first, liberate from scale second. I guess they commute ... but: To be sure we've been talking about the same category Euc of what France called Euclid's spaces, for the Euc I have in mind, Euc(E_m,E_n) has (m+1)n - (m+1 choose 2) + 1 degrees of freedom when 1 <= m <= n. Outside that range Euc(E_m,E_n) has n degrees of freedom, with rigidity obliging E_m for m > n to collapse to E_0. When m = n > 0 this reduces to 1 + n + (n choose 2), corresponding to the one dilatation, n translations, and rotation group of order (n choose 2) = n(n-1)/2 in E_n. Here's a table. . 0 1 2 3 4 5 6 7 8 9 ----------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 1 | 0 2 4 6 8 10 12 14 16 18 2 | 0 1 4 7 10 13 16 19 22 25 3 | 0 1 2 7 11 15 19 23 27 31 4 | 0 1 2 3 11 16 21 26 31 36 5 | 0 1 2 3 4 16 22 28 34 40 6 | 0 1 2 3 4 5 22 29 36 43 7 | 0 1 2 3 4 5 6 29 37 45 8 | 0 1 2 3 4 5 6 7 37 46 9 | 0 1 2 3 4 5 6 7 8 46 In contrast Aff_R(A_m,A_n) (affine spaces) has (m+1)n degrees of freedom, while Vct_R(V_m,V_n) (vector spaces) has mn. If Euler invented affine geometry as being somehow cleaner than Euclidean geometry he was going in the right direction, with vector spaces making things simpler yet via familiar mxn matrices. Regarding Bob Knighten's question on approaches to synthetic Euclidean geometry such as Hilbert's axiomatization, has anyone axiomatized more than E_3 (Euclid Books 11-13)? A satisfactory definition of Euclidean space needs to account for all finite dimensions. Vaughan